Elliptic Hypergeometric Summations by Taylor Series Expansion and Interpolation
We use elliptic Taylor series expansions and interpolation to deduce a number of summations for elliptic hypergeometric series. We extend to the well-poised elliptic case results that in the q-case have previously been obtained by Cooper and by Ismail and Stanton. We also provide identities involvin...
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| description | We use elliptic Taylor series expansions and interpolation to deduce a number of summations for elliptic hypergeometric series. We extend to the well-poised elliptic case results that in the q-case have previously been obtained by Cooper and by Ismail and Stanton. We also provide identities involving S. Bhargava's cubic theta functions.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 12 (2016), 039, 21 pages
Elliptic Hypergeometric Summations
by Taylor Series Expansion and Interpolation?
Michael J. SCHLOSSER and Meesue YOO
Fakultät für Mathematik, Universität Wien,
Oskar-Morgenstern-Platz 1, A-1090 Vienna, Austria
E-mail: michael.schlosser@univie.ac.at, meesue.yoo@univie.ac.at
URL: http://www.mat.univie.ac.at/~schlosse/
Received March 01, 2016, in final form April 13, 2016; Published online April 19, 2016
http://dx.doi.org/10.3842/SIGMA.2016.039
Abstract. We use elliptic Taylor series expansions and interpolation to deduce a number
of summations for elliptic hypergeometric series. We extend to the well-poised elliptic case
results that in the q-case have previously been obtained by Cooper and by Ismail and
Stanton. We also provide identities involving S. Bhargava’s cubic theta functions.
Key words: elliptic hypergeometric series; summations; Taylor series expansion; interpola-
tion
2010 Mathematics Subject Classification: 30E05; 33D15; 33D70; 33E05; 33E20
1 Introduction
Previously, one of us [21] established an elliptic Taylor expansion theorem which extends Is-
mail’s [11] expansion for functions symmetric in z and 1/z in terms of the Askey–Wilson mono-
mial basis. The expansion theorem in [21] involves a special case of Rains’ [17] elliptic extension
of the Askey–Wilson divided difference operator. As applications, new simple proofs were given
for Frenkel and Turaev’s [8] elliptic extensions of Jackson’s 8φ7 summation and of Bailey’s 10φ9
transformation. A further application concerned the computation of the connection coefficients
of Spiridonov’s [23] elliptic extension of Rahman’s biorthogonal rational functions.
Here we take a closer look at elliptic Taylor expansions. In particular, we describe the action
of the m-th elliptic divided difference on a function, expressed in terms of the function. In the
ordinary case, if δh denotes the central difference operator, defined by δh f(x) = f(x + h
2 ) −
f(x− h
2 ), the m-th difference is given by
δmh f(x) =
m∑
k=0
(−1)k
(
m
k
)
f
(
x+
(m
2
− k
)
h
)
.
For the q-case, where δh is replaced by the Askey–Wilson operator Dq, acting on functions f(z)
symmetric in z and 1/z, an explicit formula for Dmq f(z) was established by Cooper [5]. One
of the results of our paper concerns an extension of Cooper’s formula to the elliptic setting.
We remark that Ismail, Rains and Stanton [12] independently have also proved an elliptic ex-
tension of Cooper’s formula which turns out to be equivalent to our result by a multiplication
of operators. In [14], Ismail and Stanton have used Cooper’s explicit formula to work out an
explicit interpolation formula for polynomials symmetric in z and 1/z. Likewise, we use our
elliptic extension of Cooper’s formula to find an elliptic interpolation formula. Application of
this formula yields single and multivariable identities of Karlsson–Minton type.
?This paper is a contribution to the Special Issue on Orthogonal Polynomials, Special Functions and Applica-
tions. The full collection is available at http://www.emis.de/journals/SIGMA/OPSFA2015.html
mailto:michael.schlosser@univie.ac.at
mailto:meesue.yoo@univie.ac.at
http://www.mat.univie.ac.at/~schlosse/
http://dx.doi.org/10.3842/SIGMA.2016.039
http://www.emis.de/journals/SIGMA/OPSFA2015.html
2 M.J. Schlosser and M. Yoo
Ismail and Stanton [13] not only considered Taylor expansions in terms of the Askey–Wilson
monomial basis {(az, a/z; q)n, n ≥ 0} (see the subsequent subsection for the q-shifted factorial
notation), but also in terms of the basis
{(
q
1
4 z, q
1
4 /z; q
1
2
)
n
, n ≥ 0
}
, for which they deduced
quadratic summations as applications. We are able to extend Ismail and Stanton’s analysis and
provide, in particular, a Taylor expansion for an elliptic extension of this other basis. We note
that in addition, Ismail and Stanton [13, Theorem 2.2] gave a Taylor expansion theorem for the
basis
{
(1 + z2)
(
−q2−nz2; q2
)
n−1z
−n, n ≥ 0
}
, however this result (which involves an evaluation
at z = 0) appears not to extend to the elliptic setting.
Finally, we consider series partially involving products of S. Bhargava’s [3] cubic theta func-
tions. Such series have not been considered before. We introduce two different cubic theta
extensions of shifted factorials which are designed in such forms that they behave well under the
iterated action of the elliptic Askey–Wilson operator. Applications of Taylor expansion yield
cubic theta extensions of Jackson’s 8φ7 summation formula and of a quadratic summation of
Gessel and Stanton.
Before we present our new results, to make this paper more self-contained, we briefly review
some important material from the theory of elliptic hypergeometric series. Afterwards we turn
to the Askey–Wilson operator and its elliptic extension, and then we provide our new results.
1.1 Elliptic hypergeometric series
For basic hypergeometric series, see Gasper and Rahman’s textbook [9]. Elliptic hypergeometric
series are treated there in Chapter 11.
By definition, a function is elliptic if it is meromorphic and doubly periodic. It is well known
(cf., e.g., [25]) that elliptic functions can be built from quotients of theta functions.
As building blocks we will use the modified Jacobi theta function with argument x and nome p,
defined (in multiplicative notation) by
θ(x; p) =
∏
j≥0
((
1− pjx
)(
1− pj+1/x
))
, θ(x1, . . . , xm; p) =
m∏
k=1
θ(xk; p),
where x, x1, . . . , xm 6= 0, |p| < 1.
The modified Jacobi theta functions satisfy the following basic properties which are essential
in the theory of elliptic hypergeometric series:
θ(x; p) = −xθ(1/x; p), (1.1a)
θ(px; p) = −1
x
θ(x; p), (1.1b)
and the addition formula
θ(xy, x/y, uv, u/v; p)− θ(xv, x/v, uy, u/y; p) = u
y
θ(yv, y/v, xu, x/u; p) (1.1c)
(cf. [26, p. 451, Example 5]).
Note that in the theta function θ(x; p) we cannot let x→ 0 (unless we first let p→ 0) for x
is a pole of infinite order.
Further, we define the theta shifted factorial (or q, p-shifted factorial) by
(a; q, p)n =
n−1∏
k=0
θ
(
aqk; p
)
, n = 1, 2, . . . ,
1, n = 0,
1/
−n−1∏
k=0
θ
(
aqn+k; p
)
, n = −1,−2, . . . ,
Elliptic Hypergeometric Summations by Taylor Series Expansion and Interpolation 3
together with
(a1, a2, . . . , am; q, p)n =
m∏
k=1
(ak; q, p)n,
for compact notation. For p = 0 we have θ(x; 0) = 1 − x and, hence, (a; q, 0)n = (a; q)n is
a q-shifted factorial in base q. The parameters q and p in (a; q, p)n are called the base and nome,
respectively. Observe that
(pa; q, p)n = (−1)na−nq−(
n
2)(a; q, p)n,
which follows from (1.1b). A list of other useful identities for manipulating the q, p-shifted
factorials is given in [9, Section 11.2].
A series
∑
cn is called an elliptic hypergeometric series if g(n) = cn+1/cn is an elliptic
function of n with n considered as a complex variable, i.e., the function g(x) is a doubly periodic
meromorphic function of the complex variable x. Without loss of generality, by the theory of
theta functions, one may assume that
g(x) =
θ
(
a1q
x, a2q
x, . . . , as+1q
x; p
)
θ
(
q1+x, b1qx, . . . , bsqx; p
) z,
where the elliptic balancing condition, namely
a1a2 · · · as+1 = qb1b2 · · · bs,
holds. If we write q = e2πiσ, p = e2πiτ , with complex σ, τ , then g(x) is indeed periodic in x with
periods σ−1 and τσ−1.
For convergence reasons, one usually requires as+1 = q−n (n being a nonnegative integer), so
that the sum of an elliptic hypergeometric series is in fact finite.
Very-well-poised elliptic hypergeometric series are defined as
s+1Vs(a1; a6, . . . , as+1; q, p) :=
∞∑
k=0
θ
(
a1q
2k; p
)
θ(a1; p)
(a1, a6, . . . , as+1; q, p)k
(q, a1q/a6, . . . , a1q/as+1; q, p)k
(qz)k,
where
q2a26a
2
7 · · · a2s+1 = (a1q)
s−5.
Note that in the elliptic case the number of pairs of numerator and denominator parameters
involved in the construction of the very-well-poised term θ(a1q
2k; p)/θ(a1; p) is four (whereas in
the basic case this number is two, in the ordinary case only one). See Spiridonov [23] or Gasper
and Rahman [9, Chapter 11] for details.
In their study of elliptic 6j symbols (which are elliptic solutions of the Yang–Baxter equation
found by Baxter [2] and Date et al. [7]), Frenkel and Turaev [8] discovered the following 10V9
summation formula (as a result of a more general 12V11 transformation, being a consequence of
the tetrahedral symmetry of the elliptic 6j symbols):
10V9
(
a; b, c, d, e, q−n; q, p
)
=
(aq, aq/bc, aq/bd, aq/cd; q, p)n
(aq/b, aq/c, aq/d, aq/bcd; q, p)n
, (1.2)
where a2qn+1 = bcde. The 10V9 summation is an elliptic analogue of Jackson’s 8φ7 summation
formula (cf. [9, equation (2.6.2)])
n∑
k=0
(1− aq2k)(a, b, c, d, e, q−n; q)k
(1− a)(q, aq/b, aq/c, aq/d, aq/e, aqn+1; q)k
qk =
(aq, aq/bc, aq/bd, aq/cd; q)n
(aq/b, aq/c, aq/d, aq/bcd; q)n
, (1.3)
where a2qn+1 = bcde, which in turn is a q-analogue of Dougall’s 7F6 summation formula.
4 M.J. Schlosser and M. Yoo
1.2 The Askey–Wilson operator
The Askey–Wilson operator Dq was first defined in [1]. We consider meromorphic functions f(z)
symmetric in z and 1/z. Writing z = eiθ (note that θ need not to be real), we may consider f
to be a function in x = cos θ = (z + 1/z)/2 and write f [x] := f(z). (I.e., f can be considered as
a function in z, or equivalently, as a function in x, where the two different notations specify the
dependency to be considered.)
The Askey–Wilson operator acts on functions of x = cos θ. It is defined as follows:
Dqf [x] =
f
(
q
1
2 z
)
− f
(
q−
1
2 z
)
ι
(
q
1
2 z
)
− ι
(
q−
1
2 z
) , (1.4)
where ι[x] = x (i.e., ι(z) = (z + 1/z)/2). Equation (1.4) can also be written as
Dqf [x] =
f
(
q
1
2 z
)
− f
(
q−
1
2 z
)
i
(
q
1
2 − q−
1
2
)
sin θ
.
The operator Dq is a q-analogue of the differentiation operator (which is different to Jackson’s
q-difference operator). In particular, since
DqTn[x] =
q
n
2 − q−
n
2
q
1
2 − q−
1
2
Un−1[x],
where Tn[cos θ] = cosnθ and Un[cos θ] = sin(n+1)θ/ sin θ are the Chebyshev polynomials of the
first and second kind, one easily sees that Dq maps polynomials to polynomials, lowering the
degree by one.
In the calculus of the Askey–Wilson operator the so-called “Askey–Wilson monomials”
φn(x; a) = (az, a/z; q)n form a natural basis for polynomials or power series in x. One readily
computes
Dq(az, a/z; q)n = −2a(1− qn)
(1− q)
(
aq
1
2 z, aq
1
2 /z; q
)
n−1.
Ismail [11] proved the following Taylor theorem for polynomials f [x].
Theorem 1.1. If f [x] is a polynomial in x of degree n, then
f [x] =
n∑
k=0
fkφk(x; a),
where
fk =
(q − 1)k
(2a)k(q; q)k
q−k(k−1)/4
[
Dkq f [x]
]
x=xk
, xk :=
1
2
(
aq
k
2 + q−
k
2 /a
)
.
As it was shown in [11], the application of Theorem 1.1 to f(z) = (bz, b/z; q)n immediately
gives the q-Pfaff–Saalschütz summation (cf. [9, equation (1.7.2)]), in the form
(bz, b/z; q)n
(ba, b/a; q)n
= 3φ2
[
az, a/z, q−n
ab, q1−na/b
; q, q
]
,
and its application to the Askey–Wilson polynomials,
ωn(x; a, b, c, d; q) := 4φ3
[
az, a/z, abcdqn−1, q−n
ab, ac, ad
; q, q
]
,
Elliptic Hypergeometric Summations by Taylor Series Expansion and Interpolation 5
gives a connection coefficient identity which, by specialization, can be reduced to the Sears
transformation (cf. [9, equation (3.2.1)]), in the form
ωn(x; a, b, c, d; q) =
an(bc, bd; q)n
bn(ac, ad; q)n
ωn(x; b, a, c, d; q).
Ismail and Stanton [13] extended the above polynomial Taylor theorem to hold for entire
functions of exponential growth, resulting in infinite Taylor expansions. Marco and Parcet [15]
extended this yet further to hold for arbitrary q-differentiable functions, resulting in infinite
Taylor expansions with explicit remainder term. Among other results they were able to recover
the nonterminating q-Pfaff–Saalschütz summation (cf. [9, Appendix (II.24)]).
1.3 The well-poised and elliptic Askey–Wilson operator
Since
Dq
(az, a/z; q)n
(cz, c/z; q)n
=
2(
q
1
2 − q−
1
2
)
(z − 1/z)
[(
aq
1
2 z, aq−
1
2 /z; q
)
n(
cq
1
2 z, cq−
1
2 /z; q
)
n
−
(
aq−
1
2 z, aq
1
2 /z; q
)
n(
cq−
1
2 z, cq
1
2 /z; q
)
n
]
=
2(
q
1
2 − q−
1
2
)
(z − 1/z)
(
aq
1
2 z, aq
1
2 /z; q
)
n−1(
cq
1
2 z, cq
1
2 /z; q
)
n−1
×
[(
1− azqn−
1
2
)(
1− aq−
1
2 /z
)(
1− czqn−
1
2
)(
1− cq−
1
2 /z
) − (1− azq− 1
2
)(
1− aqn−
1
2 /z
)(
1− czq−
1
2
)(
1− cqn−
1
2 /z
) ]
=
(−1)2a(1− c/a)
(
1− acqn−1
)
(1− qn)(
1− czq−
1
2
)(
1− czq
1
2
)(
1− cq−
1
2 /z
)(
1− cq
1
2 /z
)
(1− q)
(
aq
1
2 z, aq
1
2 /z; q
)
n−1(
cq
3
2 z, cq
3
2 /z; q
)
n−1
,
we were led in [21] to define a c-generalized well-poised Askey–Wilson operator acting on x
(or z) by
Dc,q =
(
1− czq−
1
2
)(
1− czq
1
2
)(
1− cq−
1
2 /z
)(
1− cq
1
2 /z
)
Dq,
which acts “degree-lowering” on the “rational monomials” (or “well-poised monomials”)
(az, a/z; q)n
(cz, c/z; q)n
in the form
Dc,q
(az, a/z; q)n
(cz, c/z; q)n
=
(−1)2a(1− c/a)
(
1− acqn−1
)
(1− qn)
(1− q)
(
aq
1
2 z, aq
1
2 /z; q
)
n−1(
cq
3
2 z, cq
3
2 /z; q
)
n−1
.
Clearly, D0,q = Dq.
More generally, for parameters c, q, p with |q|, |p| < 1, we defined an elliptic extension of the
Askey–Wilson operator, acting on functions symmetric in z±1, by
Dc,q,pf(z) = 2q
1
2 z
θ
(
czq−
1
2 , czq
1
2 , cq−
1
2 /z, cq
1
2 /z; p
)
θ(q, z2; p)
(
f
(
q
1
2 z
)
− f
(
q−
1
2 z
))
. (1.5)
Note that Dc,q,0 = Dc,q.
In particular, using (1.1c), we have
Dc,q,p
(az, a/z; q, p)n
(cz, c/z; q, p)n
=
(−1)2aθ
(
c/a, acqn−1, qn; p
)
θ(q; p)
(
aq
1
2 z, aq
1
2 /z; q, p
)
n−1(
cq
3
2 z, cq
3
2 /z; q, p
)
n−1
. (1.6)
6 M.J. Schlosser and M. Yoo
Remark 1.2. The operator Dc,q,p happens to be a special case of a multivariable difference
operator introduced by Rains in [16]. Already in the single variable case Rains’ operator involves
two more parameters than Dc,q,p. (Rains’ difference operators generate a representation of the
Sklyanin algebra, as observed in [16] and made explicit in [18] and [19, Section 6].) Rains’
operator can be specialized to act as degree-lowering (as the above Dc,q,p does), degree-preserving
or degree-raising on abelian functions. Rains used his multivariable difference operators in [16]
to construct BCn-symmetric biorthogonal abelian functions which generalize Koornwinder’s
orthogonal polynomials. He further used his operator in [17] to derive BCn-symmetric extensions
of Frenkel and Turaev’s 10V9 summation and 12V11 transformation.
2 Elliptic Taylor expansions and interpolation
We work in the following space of abelian functions.
For a complex number c, let
Wn
c := spanC
{
gk(z)
(cz, c/z; q, p)k
, 0 ≤ k ≤ n
}
,
where gk(z) runs over all functions being holomorphic for z 6= 0 with gk(z) = gk(1/z) and
gk(pz) =
1
pkz2k
gk(z).
In classical terminology, gk(z) is an even theta function of order 2k and zero characteristic.
Rains [17] refers to such functions as BC1 theta functions of degree k, whereas in Rosengren and
Schlosser [20] they are referred to as Dk theta functions. It is well-known that the space V k of
even theta functions of order 2k and zero characteristic has dimension k+1 (see, e.g., Weber [25,
p. 49]).
Note that Wn
c consists of certain abelian functions. (For p → 0 these degenerate to certain
rational functions which we may call “well-poised”.)
Lemma 2.1 ([21, Lemma 4.1]). For any arbitrary but fixed complex number a (satisfying a 6=
cqjpk, for j = 0, . . . , n − 1, and k ∈ Z, and a 6= qjpk/c, for j = 2 − 2n, . . . , 1 − n, and k ∈ Z),
the set{
(az, a/z; q, p)k
(cz, c/z; q, p)k
, 0 ≤ k ≤ n
}
forms a basis for Wn
c .
Note that, in view of (1.6), the elliptic Askey–Wilson operator maps functions in Wn
c to
functions in Wn−1
cq
3
2
.
We now define
D(k)
c,q,p = D
(k−1)
cq
3
2 ,q,p
Dc,q,p,
with D(0)
c,q,p = ε, the identity operator. We have the following elliptic Taylor expansion theorem
which extends Theorem 1.1 of Ismail.
Theorem 2.2 ([21, Theorem 4.2]). If f is in Wn
c , then
f(z) =
n∑
k=0
fk
(az, a/z; q, p)k
(cz, c/z; q, p)k
,
Elliptic Hypergeometric Summations by Taylor Series Expansion and Interpolation 7
where
fk =
(−1)kq−k(k−1)/4θ(q; p)k
(2a)k(q, c/a, acqk−1; q, p)k
[
D(k)
c,q,pf(z)
]
z=aq
k
2
.
Example 2.3. Let
f(z) =
(bz, b/z; q, p)n
(cz, c/z; q, p)n
.
Application of Theorem 2.2 in conjunction with (1.6) gives
fk =
(−1)kq−k(k−1)/4θ(q; p)k
(2a)k(q, c/a, acqk−1; q, p)k
× (−1)k(2b)kqk(k−1)/4
(q; q, p)n
(
c/b, bcqn−1; q, p
)
k
(q; q, p)n−kθ(q; p)k
(
abqk, b/a; q, p
)
n−k(
acq2k, cqk/a; q, p
)
n−k
=
(ab, b/a; q, p)n
(ac, c/a; q, p)n
θ
(
acq2k−1; p
)
θ
(
acq−1; p
) (acq−1, c/b, bcqn−1, q−n; q, p)k(
q, ab, aq1−n/b, acqn; q, p
)
k
qk,
thus yielding Frenkel and Turaev’s 10V9 summation (1.2), in the form
(ac, c/a, bz, b/z; q, p)n
(ab, b/a, cz, c/z; q, p)n
= 10V9
(
acq−1; az, a/z, c/b, bcqn−1, q−n; q, p
)
.
We now prove an elliptic extension of a theorem of S. Cooper [5] which explicitly describes
the action of the m-iterated Askey–Wilson operator.
Theorem 2.4. The action of D(m)
c,q,p on a function f ∈Wn
c is given by
D(m)
c,q,pf(z) = (−2z)mq
m(3−m)
4
(
cq
m
2
−1z, cq
m
2
−1/z; q, p
)
m+1
(θ(q; p))m
(2.1)
×
m∑
k=0
qk(m−k)
[
m
k
]
p,q
z2(k−m)
(
cq
m
2
−kz, cq−
m
2
+k/z; q, p
)
m−1(
qm−2k+1z2; q, p
)
k
(
q2k−m+1z−2; q, p
)
m−k
f
(
q
m
2
−kz
)
,
where[
m
k
]
p,q
=
(
q1+k; q, p
)
m−k
(q; q, p)m−k
.
Proof. We prove this by induction. If m = 1, then (2.1) just reduces to the definition of
Dc,q,pf(z) in (1.5). Now say (2.1) holds up to some m. Then if we let f (m)(z) := D(m)
c,q,pf(z),
D(m+1)
c,q,p f(z) = D
cq
3
2m,q,p
f (m)(z)
= 2q
1
2 z
(
czq
3
2
m− 1
2 , cq
3
2
m− 1
2 /z; q, p
)
2
θ(q, z2; p)
(
f (m)
(
q
1
2 z
)
− f (m)
(
q−
1
2 z
))
= 2q
1
2 z
(
czq
3
2
m− 1
2 , cq
3
2
m− 1
2 /z; q, p
)
2
θ(q, z2; p)
(−2z)mq
m(3−m)
4
(θ(q; p))m
{
q
m
2
(
cq
m
2
− 1
2 z, cq
m
2
− 3
2 /z; q, p
)
m+1
×
m∑
k=0
q(k−1)(m−k)
[
m
k
]
p,q
z2(k−m)
(
cq
m
2
−k+ 1
2 z, cq−
m
2
+k− 1
2 /z; q, p
)
m−1(
qm−2k+2z2; q, p
)
k
(
q2k−mz−2; q, p
)
m−k
f
(
q
m
2
−k+ 1
2 z
)
8 M.J. Schlosser and M. Yoo
− q−
m
2
(
cq
m
2
− 3
2 z, cq
m
2
− 1
2 /z; q, p
)
m+1
×
m∑
k=0
q(k+1)(m−k)
[
m
k
]
p,q
z2(k−m)
(
cq
m
2
−k− 1
2 z, cq−
m
2
+k+ 1
2 /z; q, p
)
m−1(
qm−2kz2; q, p
)
k
(
q2k−m+2z−2; q, p
)
m−k
f
(
q
m
2
−k− 1
2 z
)}
= (2z)m+1q
m(3−m)
4
+m+1
2 (−1)m
(
czq
3
2
m− 1
2 , cq
3
2
m− 1
2 /z; q, p
)
2
(θ(q; p))m+1θ(z2; p)
(
cq
m
2
− 1
2 z, cq
m
2
− 1
2 /z; q, p
)
m
×
{
θ
(
cq
3
2
m− 1
2 z, cq
m
2
− 3
2 /z; p
)
×
m∑
k=0
q(k−1)(m−k)
[
m
k
]
p,q
z2(k−m)
(
cq
m
2
−k+ 1
2 z, cq−
m
2
+k− 1
2 /z; q, p
)
m−1(
qm−2k+2z2; q, p
)
k
(
q2k−mz−2; q, p
)
m−k
f
(
q
m
2
−k+ 1
2 z
)
− q−mθ
(
cq
m
2
− 3
2 z, cq
3
2
m− 1
2 /z; q, p
)
×
m+1∑
k=1
qk(m−k+1)
[
m
k − 1
]
p,q
z2(k−m−1)
(
cq
m
2
−k+ 1
2 z, cq−
m
2
+k− 1
2 /z; q, p
)
m−1(
qm−2k+2z2; q, p
)
k−1
(
q2k−mz−2; q, p
)
m−k+1
f
(
q
m
2
−k+ 1
2 z
)}
= (2z)m+1q
m(3−m)
4
+m+1
2 (−1)m
(
cq
m
2
− 1
2 z, cq
m
2
− 1
2 /z; q, p
)
m+2
(θ(q; p))m+1θ
(
z2; p
)
×
{
θ
(
cq
3
2
m− 1
2 z, cq
m
2
− 3
2 /z; p
)
×
q−mz−2m
(
cq
m
2
+ 1
2 z, cq−
m
2
− 1
2 /z; q, p
)
m−1(
q−mz−2; q, p
)
n
f
(
q
m
2
+ 1
2 z
)
− q−mθ
(
cq
m
2
− 3
2 z, cq
3
2
m− 1
2 /z; p
)
×
(
cq−
m
2
− 1
2 z, cq
m
2
+ 1
2 /z; q, p
)
m−1(
q−mz2; q, p
)
m
f
(
q−
m
2
− 1
2 z
)
+ q−m
m∑
k=1
qk(m−k+1)
(
q1+k; q, p
)
m−k
(q; q, p)m−k+1
(
cq
m
2
−k+ 1
2 z, cq−
m
2
+k− 1
2 /z; q, p
)
m−1f
(
q
m
2
−k+ 1
2 z
)(
qm−2k+2z2; q, p
)
k
(
q2k−mz−2; q, p
)
m−k+1
× z2k−2m
[
θ
(
qm−k+1, cq
3
2
m− 1
2 z, cq
m
2
− 3
2 /z, qkz−2; p
)
− z−2θ
(
qk, cq
m
2
− 3
2 z, cq
3
2
m− 1
2 /z, qm−k+1z2; p
)]}
= (−2z)m+1q
(m+1)(2−m)
4
(
cq
m
2
− 1
2 z, cq
m
2
− 1
2 /z; q, p
)
m+2
(θ(q; p))m+1
×
m+1∑
k=0
qk(m+1−k)
[
m+ 1
k
]
p,q
z2(k−m−1)
(
cq
m+1
2
−kz, cq−
m+1
2
+k/z; q, p
)
m(
qm−2k+2z2; q, p
)
k
(
q2k−mz−2; q, p
)
m+1−k
f
(
q
m+1
2
−kz
)
.
Hence the theorem is proved. Notice that in the last step we used the addition formula (1.1c)
with the substitutions
(x, y, u, v) 7→
(
cqm−1, q
m+1
2 z, q
m+1
2 /z, q
m+1
2
−kz
)
to simplify the summand. �
Remark 2.5. Our formula in Theorem 2.4 involves the c-dependent elliptic Askey–Wilson ope-
rator D(m)
c,q,p. Independently, Ismail, Rains and Stanton [12, Proposition 8.1] also gave an elliptic
extension of Cooper’s result. Although at first glance Ismail, Rains and Stanton’s result, which
involves a slightly different operator (without the denominator variable c), looks different from
ours in Theorem 2.4, the two results are indeed equivalent up to a multiplication of operators.
(We would like to thank Eric Rains for helping us to clarify this.) In fact, the operator(
q
m
2
−1cz, q
m
2
−1c/z; q, p
)−1
m+1
D(m)
c,q,p(cz, c/z; q, p)
−1
m−1
Elliptic Hypergeometric Summations by Taylor Series Expansion and Interpolation 9
(which acts on functions h(z) = g(z)/(czqm−1, cqm−1/z; q, p)n−m+1 with g(z) = g(1/z) and
g(pz) = p−nz−2ng(z)) is independent of c, thus our operator D(m)
c,q,p is proportional to
v2mθ
(
q
m
2 vz, q
m
2 v/z; q
)(
q
m
2
−1cz, q
m
2
−1c/z; q, p
)
m+1
Dm(q; p)(cz, c/z; q, p)m−1θ(vz, v/z; q)−1,
with Dm(q; p) from [12, Section 8], where the constant of proportionality is independent of v.
We are now able to obtain an elliptic extension of Ismail and Stanton’s [14, Theorem 3.4]
interpolation formula. In particular, for any a ∈ C, the function f ∈Wn
c is uniquely determined
by its evaluation at the n+ 1 interpolation points a, aq, . . . , aqn, with closed form coefficients.
Theorem 2.6. If f is in Wn
c , then(
a2q, q, cz, c/z; q, p
)
n
(ac, c/a, aqz, aq/z; q, p)n
f(z)
=
n∑
k=0
qk
θ
(
a2q2k; p
)
θ(a2; p)
(
q−n, a2, aq/c, acqn, az, a/z; q, p
)
k(
q, a2qn+1, ac, aq1−n/c, aqz, aq/z; q, p
)
k
f
(
aqk
)
.
Proof. By combining Theorem 2.2 and Theorem 2.4, we find that
f(z) =
n∑
k=0
qkθ
(
cq−1/a, acq2k−1; p
)(az, a/z; q, p)k
(cz, c/z; q, p)k
×
k∑
j=0
q−(k−j)
2
a2(j−k)
(
acqk−j , cq−k+j/a; q, p
)
k−1(
q, a2q2k−2j+1; q, p
)
j
(
q, a−2q2j−2k+1; q, p
)
k−j
f
(
aqk−j
)
.
By shifting the index k 7→ k + j, we get
f(z) =
n∑
k=0
n−k∑
j=0
qk+jθ
(
cq−1/a, acq2k+2j−1; p
)(az, a/z; q, p)k+j
(cz, c/z; q, p)k+j
× q−k2a−2k
(
acqk, cq−k/a; q, p
)
k+j−1(
q, a2q2k+1; q, p
)
j
(
q, a−2q−2k+1; q, p
)
k
f
(
aqk
)
=
n∑
k=0
qk(1−k)a−2k
(
acqk, cq−k/a, az, a/z; q, p
)
k(
q, a−2q−2k+1, cz, c/z; q, p
)
k
f
(
aqk
)
×
n−k∑
j=0
qj
θ
(
acq2k+2j−1; p
)
θ
(
acq2k−1; p
) (
acq2k−1, cq−1/a, aqkz, aqk/z, q−n+k, acqn+k; q, p
)
j(
q, a2q2k+1, cqk/z, cqkz, q−n+k, acqn+k; q, p
)
j
=
n∑
k=0
qk(1−k)a−2k
×
(
acqk, cq−k/a, az, a/z; q, p
)
k
(
acq2k, aqk+1/z, aqk+1z, c/a; q, p
)
n−k(
q, a−2q−2k+1, cz, c/z; q, p
)
k
(
q, a2q2k+1, cqk/z, cqkz; q, p
)
n−k
f
(
aqk
)
.
The last sum was obtained by virtue of the Frenkel and Turaev summation formula (1.2). The
theorem then follows by elementary manipulations. �
Corollary 2.7. We have the elliptic Karlsson–Minton type identity(
q, a2q; q, p
)
n
(aqz, aq/z; q, p)n
(bz, b/z; q, p)s(dz, d/z; q, p)n−s
(ab, b/a; q, p)s(ad, d/a; q, p)n−s
= 12V11
(
a2, q−n, az, a/z, aq/b, aq/d, abqs, adqn−s; q, p
)
.
10 M.J. Schlosser and M. Yoo
Proof. We apply Theorem 2.6 to
f(z) =
(bz, b/z; q, p)s(dz, d/z; q, p)n−s
(cz, c/z; q, p)n
. �
More generally, we have the following result.
Corollary 2.8. We have the elliptic Karlsson–Minton type identity(
a2q, q; q, p
)
n
(aqz, aq/z; q, p)n
n∏
j=1
θ(bjz, bj/z; p)
=
n∑
k=0
qk(n+1) θ
(
a2q2k; p
)
θ
(
a2; p
) (
q−n, a2, az, a/z; q, p
)
k(
q, a2qn+1, aqz, aq/z; q, p
)
k
n∏
j=1
θ
(
abkq
k, bjq
−k/a; p
)
.
Proof. We take
f(z) =
n∏
j=1
θ(bjz, bj/z; p)
(cz, c/z; q, p)n
(2.2)
and apply Theorem 2.6. �
Remark 2.9. It should be noted that if in the proof of Corollary 2.8 we instead would have
taken
f(z) =
t∏
j=1
θ(bjz, bj/z; p)
(cz, c/z; q, p)t
for 0 ≤ t ≤ n, we would have just obtained the special case of Corollary 2.8 with bj 7→ cqj−1 for
t+ 1 ≤ n, which is clear carrying out those specializations in (2.2).
We now consider a multivariate version of Theorem 2.6. Let us consider the space of functions
Wn1,...,nm
c1,...,cm := SpanC
{
gk1,...,km(z1, . . . , zm)
m∏
i=1
(cizi, ci/zi; q, p)ki
, 0 ≤ ki ≤ ni, i = 1, . . . ,m
}
,
where gk1,...,km(z1, . . . , zm) runs over all functions being holomorphic in z1, z2, . . . , zm 6= 0 and
symmetric in zi and 1/zi, and
gk1,...,km(z1, . . . , pzi, . . . , zm) =
1
pkiz2kii
gk1,...,km(z1, . . . , zi, . . . , zm),
for all i = 1, . . . ,m.
We define a multivariate extension of the elliptic Askey–Wilson operator as follows.
Dci,q,p;zif(z1, . . . , zm) = 2q
1
2 zi
θ
(
ciziq
− 1
2 , ciziq
1
2 , ciq
− 1
2 /zi, ciq
1
2 /zi; p
)
θ
(
q, z2i ; p
)
×
(
f
(
z1, . . . , q
1
2 zi, . . . , zm
)
− f
(
z1, . . . , q
− 1
2 zi, . . . , zm
))
,
D(k+1)
ci,q,p;zi = Dciq 3
2 k,q,p;zi
D(k)
ci,q,p;zi ,
and for c = (c1, . . . , cm), k = (k1, . . . , km), and z = (z1, . . . , zm),
D(k)
c,q,p;z = D(k1)
c1,q,p;z1 · · · D
(km)
cm,q,p;zm .
Elliptic Hypergeometric Summations by Taylor Series Expansion and Interpolation 11
Theorem 2.10. If f(z1, . . . , zm) is in Wn
c , then
f(z1, . . . , zm) =
n1,...,nm∑
k1,...,km=0
fk1,...,km
m∏
i=1
(aizi, ai/zi; q, p)ki
(cizi, ci/zi; q, p)ki
, (2.3)
where
fk1,...,km =
m∏
i=1
(−1)kiq−
ki(ki−1)
4 (θ(q; p))ki
(2ai)ki
(
q, ci/ai, aiciqki−1; q, p
)
ki
[
D(k)
c,q,p;zf(z1, . . . , zm)
]
zi=aiqki/2
,
where k = (k1, . . . , km).
Proof. Note that, for each j = 1, . . . ,m,
Dcj ,q,p;zj
m∏
i=1
(aizi, ai/zi; q, p)ni
(cizi, ci/zi; q, p)ni
(2.4)
=
(−1)2ajθ
(
cj/aj , ajcjq
nj−1, qnj ; p
)
θ(q; p)
(
ajq
1
2 zj , ajq
1
2 /zj ; q, p
)
nj−1(
cjq
3
2 zj , cjq
3
2 /zj ; q, p
)
nj−1
m∏
i=1,
i 6=j
(aizi, ai/zi; q, p)ni
(cizi, ci/zi; q, p)ni
.
Iterating (2.4) gives[
D(k)
c,q,p;zf(z1, . . . , zm)
]
zi=aiqki/2
=
m∏
i=1
(−1)ki(2ai)kiq
ki(ki−1)
4
(q; q, p)ni
(
ci/ai, aiciq
ni−1; q, p
)
ki
(q; q, p)ni−kiθ(q; p)
ki
×
(aiq ki
2 zi, aiq
ki
2 /zi; q, p
)
ni−ki(
ciq
3ki
2 zi, ciq
3ki
2 /zi; q, p
)
ni−ki
zi=aiq
ki
2
=
m∏
i=1
(−1)ki(2ai)kiq
ki(ki−1)
4
(
q, ci/ai, aiciq
ki−1; q, p
)
ki
θ(q; p)ki
δniki .
Then the theorem follows by applying D(j)
c,q,p;z to both sides of (2.3) and then setting zi = aiq
ji/2,
for i = 1, . . . ,m and j = (j1, . . . , jm). �
Now we provide a multivariate extension of Theorem 2.4.
Theorem 2.11. For n = (n1, . . . , nm), c = (c1, . . . , cm),
D(n)
c,q,p;zf(z1, . . . , zm) =
m∏
i=1
(−2zi)niq
ni(3−ni)
4
(
ciq
ni
2
−1zi, ciq
ni
2
−1/zi; q, p
)
ni+1
(θ(q; p))ni
×
nm∑
km=0
· · ·
n1∑
k1=0
m∏
i=1
(
qki(ni−ki)
[
ni
ki
]
p,q
z
2(ki−ni)
i
(
ciq
ni
2
−kizi, ciq
−ni
2
+ki/zi; q, p
)
ni−1(
qni−2ki+1z2i ; q, p
)
ki
(
q2ki−ni+1z−2i ; q, p
)
ni−ki
× f
(
q
n1
2
−k1z1, . . . , q
nm
2
−kmzm
))
.
Proof. The theorem follows by applying Theorem 2.4 successively for each i = 1, . . . ,m. �
12 M.J. Schlosser and M. Yoo
We combine Theorems 2.10 and 2.11 to obtain the following multivariable elliptic interpola-
tion formula.
Theorem 2.12. For f(z1, . . . , zm) in Wn
c , we have
m∏
i=1
(
a2i q, q, cizi, ci/zi; q, p
)
ni
(aici, ci/ai, aiqzi, aiq/zi; q, p)ni
f(z1, . . . , zm)
=
n1,...,nm∑
k1,...,km=0
m∏
i=1
qki
θ
(
a2i q
2ki ; p
)
θ
(
a2i ; p
) (
q−ni , a2i , aiq/ci, aiciq
ni , aizi, ai/zi; q, p
)
ki(
q, a2i q
ni+1, aici, aiq1−ni/ci, aiqzi, aiq/zi; q, p
)
ki
× f
(
a1q
k1 , . . . , amq
km
)
.
This theorem extends a result given by Ismail and Stanton [14, Theorem 3.10], which can be
obtained by taking m = 2, p→ 0, c1 = c2 = 0 and n1 = n2 = n.
Corollary 2.13. We have the following multivariable elliptic Karlsson–Minton type identity
m∏
i=1
(
a2i q, q; q, p
)
ni
(aiqzi, aiq/zi; q, p)ni
si∏
j=1
(bijzi, bij/zi; q, p)vij
(bijai, bij/ai; q, p)vij
∏
1≤i<j≤m
(
a
wij
i z
−wij
i θ(zizj , zi/zj ; p)
wij
×
rij∏
lij=1
(αlijzizj , αlijzi/zj , αlijzj/zi, αlij/zizj ; q, p)ulij
(αlijaiaj , αlijai/aj , αlijaj/ai, αlij/aiaj ; q, p)ulij
)
=
n1,...,nm∑
k1,...,km=0
m∏
i=1
(
qki
θ
(
a2i q
2ki ; p
)
θ
(
a2i ; p
) (
q−ni , a2i , aizi, ai/zi; q, p
)
ki(
q, a2i q
ni+1, aiqzi, aiq/zi; q, p
)
ki
×
si∏
j=1
(
aibijq
vij , aiq/bij ; q, p
)
ki(
aibij , aiq1−vij/bij ; q, p
)
ki
)
×
∏
1≤i<j≤m
rij∏
lij=1
q
−2ulij ki
×
(
αlijaiajq
ulij , qaiaj/αlij ; q, p
)
ki+kj
(
αlijaiq
ulij /aj , qai/ajαlij ; q, p
)
ki−kj(
αlijaiaj , q
1−ulij aiaj/αlij ; q, p
)
ki+kj
(
αlijai/aj , q
1−ulij ai/ajαlij ; q, p
)
ki−kj
×
∏
1≤i<j≤m
q−wijkiθ
(
aiajq
ki+kj , aiq
ki−kj/aj ; p
)wij ,
where
ni =
si∑
j=1
vij +
i−1∑
j=1
wji +
m∑
j=i+1
wij + 2
rij∑
l=1
i−1∑
j=1
ulji +
m∑
j=i+1
ulij
,
for i = 1, . . . ,m.
Proof. We apply Theorem 2.12 to
f(z1, . . . , zm) =
m∏
i=1
si∏
j=1
(bijzi, bij/zi; q, p)vij
(cizi, ci/zi; q, p)ni
∏
1≤i<j≤m
z
−wij
i θ(zizj , zi/zj ; p)
wij
×
∏
1≤i<j≤m
rij∏
lij=1
(αlijzizj , αlijzi/zj , αlijzj/zi, αlij/zizj ; q, p)ulij ,
Elliptic Hypergeometric Summations by Taylor Series Expansion and Interpolation 13
where
ni =
si∑
j=1
vij +
i−1∑
j=1
wji +
m∑
j=i+1
wij + 2
i−1∑
j=1
rji∑
lji=1
ulji + 2
m∑
j=i+1
rij∑
lij=1
ulij ,
for i = 1, . . . ,m. �
Corollary 2.13 extends a result by Ismail and Stanton (see [14, Corollary 3.11]), corresponding
to a special case of its m = 2 instance.
More generally, f(z1, . . . , zm) could involve symmetrized products of 2k factors of the form
(λz±i1z
±
i2
· · · z±ik ; q, p)y (the notation z±ij means that the resprective variable could appear as zij
or z−1ij , where all possible combinations appear), where {i1, . . . , ik} is any subset of {1, . . . , n}.
(In the corollary, we only considered factors for k = 1, 2.)
Corollary 2.13 can be easily seen to be equivalent to its ulij = 1 and vij = 1 case, for all i, j, in
which case the respective factorials reduce to simple theta functions. To recover the general case
from this special case one can suitably increase rij and s1, . . . , sm and choose the parameters
partially in geometric progression to obtain shifted factorials. In particular, we can replace rij
by u1 + · · · + urij and relabel αu1+···+ulij−1+h 7→ αlijq
h−1, for all 1 ≤ lij ≤ rij , 1 ≤ h ≤ ulij ,
etc. (One could even add extra bases, in addition to q. This feature is typical for series of
Karlsson–Minton type.)
For convenience, we restate the corollary in this equivalent form.
Corollary 2.14. We have the following multivariable elliptic Karlsson–Minton type identity
m∏
i=1
(a2i q, q; q, p)ni
(aiqzi, aiq/zi; q, p)ni
si∏
j=1
θ(bijzi, bij/zi; p)
θ(bijai, bij/ai; p)
×
∏
1≤i<j≤m
awij
i z
−wij
i θ(zizj , zi/zj ; p)
wij
rij∏
lij=1
θ(αlijzizj , αlijzi/zj , αlijzj/zi, αlij/zizj ; p)
θ(αlijaiaj , αlijai/aj , αlijaj/ai, αlij/aiaj ; p)
=
n1,...,nm∑
k1,...,km=0
m∏
i=1
qki θ(a2i q2ki ; p)
θ(a2i ; p)
(q−ni , a2i , aizi, ai/zi; q, p)ki
(q, a2i q
ni+1, aiqzi, aiq/zi; q, p)ki
si∏
j=1
θ(aibijq
ki , aiq
ki/bij ; p)
θ(aibij , ai/bij ; p)
×
∏
1≤i<j≤m
rij∏
lij=1
q−2ki
θ
(
αlijaiajq
ki+kj , qki+kjaiaj/αlij , αlijaiq
ki−kj/aj , q
ki−kjai/ajαlij ; p
)
θ(αlijaiaj , aiaj/αlij , αlijai/aj , ai/ajαlij ; p)
×
∏
1≤i<j≤m
q−wijkiθ
(
aiajq
ki+kj , aiq
ki−kj/aj ; p
)wij ,
where
ni = si +
i−1∑
j=1
wji +
m∑
j=i+1
wij + 2
i−1∑
j=1
rji + 2
m∑
j=i+1
rij ,
for i = 1, . . . ,m.
2.1 A quadratic elliptic Taylor expansion theorem
In [13] Ismail and Stanton also considered the basis {φk(z), 0 ≤ k ≤ n} where φk(z) =(
q1/4z, q1/4/z; q1/2
)
k
. The set{(
q1/4z, q1/4/z; q1/2, p
)
k
(cz, c/z; q, p)k
, 0 ≤ k ≤ n
}
14 M.J. Schlosser and M. Yoo
apparently forms a basis for Wn
c . We now provide a Taylor expansion theorem with respect to
this basis.
Theorem 2.15. If f is in Wn
c , then
f(z) =
n∑
k=0
fk
(
q1/4z, q1/4/z; q1/2, p
)
k
(cz, c/z; q, p)k
, (2.5)
where
fk =
(−1)kq−k/4θ(q; p)k
2k(q; q, p)k
(
cq
k
2
− 3
4 ; q1/2, p
)
2k
[
D(k)
c,q,pf(z)
]
z=q1/4
.
Proof. Note that
D(k)
c,q,p
((
q1/4z, q1/4/z; q1/2, p
)
n
(cz, c/z; q, p)n
)
=
(−2)kqk/4
(
cq
n
2
− 3
4 ; q1/2, p
)
2k
(q; q, p)n
(q; q, p)n−kθ(q; p)k
(
q1/4z, q1/4/z; q1/2, p
)
n−k(
cq
3
2
kz, cq
3
2
k/z; q, p
)
n−k
,
which can be proved by induction. The theorem then follows by applying D(j)
c,q,p to both sides
of (2.5) and then setting z = q1/4. �
In the following, we recover an elliptic quadratic summation by Warnaar [24, Corollary 4.4;
b = a], which was originally proved by using inverse relations. Its p = 0 case has been given
earlier by Gessel and Stanton [10, equation (1.4)].
Corollary 2.16. We have the following summation
(az, a/z; q, p)n
(cz, c/z; q, p)n
(
cq−1/4; q1/2, p
)
2n(
aq−1/4; q1/2, p
)
2n
=
n∑
k=0
q
k
2
θ
(
cq
3
2
k− 3
4 ; p
)
θ
(
cq−
3
4 ; p
) (c/a, acqn−1, q−n; q, p)k
(cz, c/z, q; q, p)k
(
cq−
3
4 , q
1
4 z, q
1
4 /z; q1/2, p
)
k(
aq−
1
4 , cqn−
1
4 , q
3
4
−n/a; q1/2, p
)
k
.
Proof. We apply Theorem 2.15 to
f(z) =
(az, a/z; q, p)n
(cz, c/z; q, p)n
. �
Remark 2.17. If we expand(
q
1
4 z, q
1
4 /z; q
1
2 , p
)
n
(cz, c/z; q, p)n
in terms of
(az, a/z; q, p)n
(cz, c/z; q, p)n
using Theorem 2.2, we obtain(
q
1
4 z, q
1
4 /z; q
1
2 , p
)
n(
aq
1
4 , q
1
4 /a; q
1
2 , p
)
n
(ac, c/a; q, p)n
(cz, c/z; q, p)n
Elliptic Hypergeometric Summations by Taylor Series Expansion and Interpolation 15
=
n∑
k=0
qk
θ
(
acq2k−1; p
)
θ
(
acq−1; p
) (q−n, acq−1, az, a/z, cq n
2
− 3
4 , cq
n
2
− 1
4 ; q, p
)
n(
q, acqn, cz, c/z, aq
1
4
−n
2 , aq
3
4
−n
2 ; q, p
)
n
. (2.6)
At first glance this appears to be a true quadratic summation formula. However, the right-hand
side of (2.6) is
10V9
(
acq−1; az, a/z, cq
n
2
− 3
4 , cq
n
2
− 1
4 , q−n; q, p
)
,
which, by Frenkel and Turaev’s summation formula (1.2), can be reduced to(
ac, c/a, q
3
4
−n
2 z, q
1
4
−n
2 /z; q, p
)
n(
cz, c/z, aq
3
4
−n
2 , q
3
4
−n
2 /a; q, p
)
n
.
Elementary manipulations can now be applied to transform this expression to the left-hand side
of (2.6).
3 Expansions involving cubic theta functions
The cubic theta function γ(z, a; p) with two independent variables z and a in addition to the
nome p was considered by S. Bhargava [3]. (For a thorough treatment of the theory of cubic
theta functions in analogy to the theory of the classical Jacobi theta functions, see [22].) It is
defined by
γ(z, a; p) =
∞∑
k=−∞
∞∑
l=−∞
pk
2+kl+l2ak+lzk−l. (3.1)
This function, up to a normalization factor
(
p2; p2
)2
∞ (independent from a and z), is almost
equal to the following product of two modified Jacobi theta functions
(
p2; p2
)2
∞θ
(
−paz; p2
)
θ
(
−pa/z; p2
)
=
∞∑
k=−∞
∞∑
l=−∞
pk
2+l2ak+lzk−l,
which differs by the factor pkl to the summand of the double series in (3.1). Because of this
additional factor pkl, the cubic theta function does not factorize into a product of two modified
Jacobi theta functions of such a simple form. In principle though, the cubic theta function
could be factorized into two modified Jacobi theta functions, but their arguments would have
nontrivial expansions in a, z, and p.
From (3.1), by replacing (k, l) by (l, k), or (k, l) by (−l,−k), respectively, we immediately
deduce the symmetries [3]
γ(1/z, a; p) = γ(z, a; p), (3.2a)
and
γ(z, 1/a; p) = γ(z, a; p). (3.2b)
Further, from (3.1), by replacing (k, l) by (k + λ + µ, l + λ), it is easy to verify that for all
integers λ and µ the following functional equation holds [3]:
γ(z, a; p) = p3λ
2+3λµ+µ2a2λ+µzµγ
(
pµ/2z, p3(2λ+µ)/2a; p
)
.
16 M.J. Schlosser and M. Yoo
In particular, we have the quasi periodicities
γ(pz, a; p) =
1
pz2
γ(z, a; p), (3.3a)
and
γ(z, p3a; p) =
1
p3a2
γ(z, a; p). (3.3b)
Further, by separating the terms in the expansion of p according to whether the exponents of p
are divisible by 3 or not, one can show [3]
γ(z, a; p) = γ
(√
az3,
√
a3/z3; p3
)
+ paz−1γ
(√
az3, p3
√
a3/z3; p3
)
,
while separating the terms in the expansion of z according to whether the exponents of z are
even or odd, one has [4]
γ(z, a; p) =
(
p6; p6
)
∞
(
p2; p2
)
∞
[
θ
(
−p3a; p6
)
θ
(
−pz2; p2
)
+ pazθ
(
−p6a2; p6
)
θ
(
−p2z2; p2
)]
.
Cooper and Toh [6] proved the following addition formulae which will be useful in our com-
putations.
Lemma 3.1 ([6, Corollary 4.5]). The following identities connecting modified Jacobi theta func-
tions and cubic theta functions hold:
γ(z1, α; p)θ(z3/z2, z2z3; p)− γ(z2, α; p)θ(z3/z1, z1z3; p)
=
z3
z1
γ(z3, α; p)θ(z1/z2, z1z2; p), (3.4a)
and
γ
(
z, α1; p
1
3
)
θ(α3/α2, α2α3; p)− γ
(
z, α2; p
1
3
)
θ(α3/α1, α1α3; p)
=
α3
α1
γ
(
z, α3; p
1
3
)
θ(α1/α2, α1α2; p). (3.4b)
These two identities were proved in [6] by specializing a (3 × 3) determinant evaluation
involving cubic theta functions. They can also be proved directly, expanding the cubic theta
functions and modified Jacobi theta functions as infinite series, together with clever series
rearrangement.
Now we introduce the first cubic theta analogue of the q-shifted factorial by
〈az, a/z; q, p〉n :=
n−1∏
j=0
γ
(
zq
1−n
2
+j , aq
n−1
2 ; p
)
.
From (3.3a) it is easy to see that the cubic shifted factorial satisfies
〈apz, a/pz; q, p〉n =
1
pnz2n
〈az, a/z; q, p〉n.
Together with (3.2a), this implies that the quotient
〈az, a/z; q, p〉n
(cz, c/z; q, p)n
is in the space Wn
c . Hence we can apply Theorem 2.2 to it, by which we obtain the first cubic
theta extension of Jackson’s 8φ7 summation (1.3).
Elliptic Hypergeometric Summations by Taylor Series Expansion and Interpolation 17
Corollary 3.2. We have the following summation
(bc, c/b; q, p)n
〈az, a/z; q, p〉n
(cz, c/z; q, p)n
=
n∑
k=0
qnk
(c
b
)k θ(bcq2k−1; p)
θ(bcq−1; p)
(q−n, bcq−1, bz, b/z; q, p)k
(q, bcqn, cz, c/z; q, p)k
×
〈
acqn−1, aq1−k/c; q, p
〉
k
〈abqk, aq−k/b; q, p〉n
〈abqn, aq−k/b; q, p〉k
. (3.5)
Proof. By using (3.4a) in Lemma 3.1, we can prove by induction that
D(k)
c,q,p
(
〈az, a/z; q, p〉n
(cz, c/z; q, p)n
)
= (2c)kq
3
4
k(k−1) (q
n; q−1, p)k
θ(q; p)k
k−1∏
j=0
γ(cq
n−1
2
+j , aq
n−1
2 ; p)
〈aq
k
2 z, aq
k
2 /z; q, p〉n−k
(cq
3
2
kz, cq
3
2
k/z; q, p)n−k
= (2c)kq
3
4
k(k−1) (q
n; q−1, p)k
θ(q; p)k
〈acqn−1, aq1−k/c; q, p〉k
〈aq
k
2 z, aq
k
2 /z; q, p〉n−k
(cq
3
2
kz, cq
3
2
k/z; q, p)n−k
.
Then the corollary follows from Theorem 2.2 while expanding in the basis
f(z) =
(bz, b/z; q, p)n
(cz, c/z; q, p)n
. �
To recover Jackson’s 8φ7 summation from Corollary 3.2, substitute
a 7→ −a
p(1 + a2qn−1)
in (3.5), multiply both sides of the identity by (1 + a2qn−1)n and let p → 0. When p → 0, the
usual theta shifted factorials clearly reduce to the q-shifted factorials. That is, the quotient on
the left-hand side reduces to
lim
p→0
(bc, c/b; q, p)n
(cz, c/z; q, p)n
=
(bc, c/b; q)n
(cz, c/z; q)n
.
What happens with the cubic theta shifted factorial? We have
lim
p→0
(
1 + a2qn−1
)n〈 −az
p(1 + a2qn−1)
,
−a
p(1 + a2qn−1)z
; p
〉
n
=
(
1 + a2qn−1
)n
lim
p→0
n−1∏
j=0
γ
(
zq
1−n
2
+j ,
−aq
n−1
2
p(1 + a2qn−1)
; p
)
=
(
1 + a2qn−1
)n n−1∏
j=0
lim
p→0
∞∑
k=−∞
∞∑
l=−∞
(−1)k+lpk2+kl+l2−k−l
(
aq
n−1
2
1 + a2qn−1
)k+l (
zq
1−n
2
+j
)k−l
.
Now it is easy to see that for p→ 0 only three terms in the various double infinite series survive.
These three terms correspond to the cases (k, l) = (0, 0), (1, 0), (0, 1). The last expression thus
reduces to(
1 + a2qn−1
)n n−1∏
j=0
(
1− aq
n−1
2
1 + a2qn−1
(
zq
1−n
2
+j + z−1q
n−1
2
−j))
=
n−1∏
j=0
(
1 + a2qn−1 − aq
n−1
2
(
zq
1−n
2
+j + z−1q
n−1
2
−j))
=
n−1∏
j=0
(1− azqj)
(
1− aqn−1−j/z
)
= (az, a/z; q)n.
We take similar limits on the right-hand side of (3.5).
18 M.J. Schlosser and M. Yoo
Our next result involves elliptic interpolation of cubic theta shifted factorials.
Corollary 3.3. We have the following Karlsson–Minton type identity involving cubic theta
functions(
a2q, q; q, p
)
n
(aqz, aq/z; q, p)n
〈bz, b/z; q, p〉n
=
n∑
k=0
qk(n+1) θ
(
a2q2k; p
)
θ
(
a2; p
) (
q−n, a2, az, a/z; q, p
)
k(
q, a2qn+1, aqz, aq/z; q, p
)
k
〈
abqk, bq−k/a; q, p
〉
n
.
Proof. We apply Theorem 2.6 to
f(z) =
〈bz, b/z; q, p〉n
(cz, c/z; q, p)n
. �
More generally, we have the following Karlsson–Minton type identity involving cubic theta
functions.
Corollary 3.4. We have(
a2q, q; q, p
)
n
(aqz, aq/z; q, p)n
s∏
i=1
θ(biz, bi/z; p)
n−s∏
j=1
γ(z, dj ; p) =
n∑
k=0
qk(n+1) θ(a
2q2k; p)
θ(a2; p)
×
(
q−n, a2, az, a/z; q, p
)
k(
q, a2qn+1, aqz, aq/z; q, p
)
k
s∏
i=1
θ
(
abiq
k, biq
−k/a; p
) n−s∏
j=1
γ
(
aqk, dj ; p
)
.
Proof. We apply Theorem 2.6 to
f(z) =
s∏
i=1
θ(biz, bi/z; p)
n−s∏
j=1
γ(z, dj ; p)
(cz, c/z; q, p)n
. �
Our next result concerns a cubic theta extension of Gessel and Stanton’s quadratic summation
[10, equation (1.4)].
Corollary 3.5. We have the following summation
〈az, a/z; q, p〉n
(cz, c/z; q, p)n
(
cq−
1
4 , cq
1
4 ; q, p
)
n
=
n∑
k=0
ckq
k
4
(k−2)+nk θ
(
cq
3
2
k− 3
4 ; p
)
θ
(
cq
k
2
− 3
4 ; p
) (q−n; q, p)k
(q; q, p)k
(
cq−
1
4 ; q
1
2 , p
)
k(
cqn−
1
4 ; q
1
2 , p
)
k
(
q
1
4 z, q
1
4 /z; q
1
2 , p
)
k
(cz, c/z; q, p)k
×
〈
acqn−1, aq1−k/c; q, p
〉
k
〈
aq
k
2
+ 1
4 , aq
k
2
− 1
4 ; q, p
〉
n−k. (3.6)
Proof. We apply Theorem 2.15 to
f(z) =
〈az, a/z; q, p〉n
(cz, c/z; q, p)n
. �
Similarly to the way we recovered Jackson’s 8φ7 summation from Corollary 3.2, Gessel and
Stanton’s quadratic summation can be readily obtained by substituting a 7→ −p−1a/
(
1+a2qn−1
)
in (3.6), multiplying both sides by (1 + a2qn−1)n and taking the limit p→ 0.
Elliptic Hypergeometric Summations by Taylor Series Expansion and Interpolation 19
Next, we define the second cubic theta shifted factorial, with base p1/3:
〈〈az, a/z; q, p
1
3 〉〉n :=
n−1∏
j=0
γ
(
aq
n−1
2 , zq
1−n
2
+j ; p
1
3
)
.
Recalling equations (3.3a) and (3.3b) (which we reformulate after interchanging a and z),
γ(a, z; p) = γ(a, 1/z; p), γ(a, z; p) = p3z2γ
(
a, p3z; p
)
,
we see that〈〈
apz, a/pz; q, p
1
3
〉〉
n
=
1
pnz2n
〈〈
az, a/z; q, p
1
3
〉〉
n
.
This implies that the quotient〈〈
az, a/z; q, p
1
3
〉〉
n
(cz, c/z; q, p)n
is also in the space Wn
c . Thus, Theorem 2.2 can be applied to it, by which we obtain the second
cubic theta extension of Jackson’s 8φ7 summation (1.3).
Corollary 3.6. We have the following summation〈〈
bz, b/z; q, p
1
3
〉〉
n
(cz, c/z; q, p)n
(ac, c/a; q, p)n =
n∑
k=0
qnk
( c
a
)k θ(acq2k−1; p)
θ
(
acq−1; p
) (q−n, acq−1, az, a/z; q, p)k(
q, acqn, cz, c/z; q, p
)
k
×
〈〈
bcqn−1, bq1−k/c; q, p
1
3
〉〉
k
〈〈
abqk, b/a; q, p
1
3
〉〉
n−k. (3.7)
Proof. Note that by using (3.4b) in Lemma 3.1, we can show by induction that
D(k)
c,q,p
(〈〈
bz, b/z; q, p
1
3
〉〉
n
(cz, c/z; q, p)n
)
= (2c)kq
3
4
k(k−1)
(
qn; q−1, p
)
k
θ(q; p)k
k−1∏
j=0
γ
(
bq
n−1
2 , cq
n−1
2
+j ; p
1
3
)〈〈bq k
2 z, bq
k
2 /z; q, p
1
3
〉〉
n−k(
cq
3
2
kz, cq
3
2
k/z; q, p
)
n−k
= (2c)kq
3
4
k(k−1)
(
qn; q−1, p
)
k
θ(q; p)k
〈〈
bcqn−1, bq1−k/c; q, p
1
3
〉〉
k
〈〈
bq
k
2 z, bq
k
2 /z; q, p
1
3
〉〉
n−k(
cq
3
2
kz, cq
3
2
k/z; q, p
)
n−k
.
Using this, we apply Theorem 2.2 to
f(z) =
〈〈
bz, b/z; q, p
1
3
〉〉
n
(cz, c/z; q, p)n
. �
To recover Jackson’s 8φ7 summation from Corollary 3.6, substitute
b 7→ −b
p
1
3
(
1 + b2qn−1
)
in (3.7), multiply both sides of the identity by
(
1 + b2qn−1
)n
and let p → 0. When p → 0,
the usual theta shifted factorials reduce to the q-shifted factorials and the cubic theta shifted
factorial on the left-hand side of (3.7) becomes
lim
p→0
(
1 + b2qn−1
)n〈〈 −bz
p
1
3
(
1 + b2qn−1
) , −b
p
1
3
(
1 + b2qn−1
)
z
; p
1
3
〉〉
n
20 M.J. Schlosser and M. Yoo
=
(
1 + b2qn−1
)n
lim
p→0
n−1∏
j=0
γ
(
−bq
n−1
2
p
1
3 (1 + b2qn−1)
, zq
1−n
2
+j ; p
1
3
)
=
(
1 + b2qn−1
)n n−1∏
j=0
lim
p→0
∞∑
k=−∞
∞∑
l=−∞
(−1)k−lp
1
3
(k2+kl+l2−k+l)
×
(
zq
1−n
2
+j
)k+l( bq
n−1
2
1 + b2qn−1
)k−l
.
Now it is easy to see that for p→ 0 only three terms in the various double infinite series survive.
These correspond to the cases (k, l) = (0, 0), (1, 0), (0,−1). The last expression thus reduces to
(
1 + b2qn−1
)n n−1∏
j=0
(
1− bq
n−1
2
1 + b2qn−1
(
zq
1−n
2
+j + z−1q
n−1
2
−j))
=
n−1∏
j=0
(
1 + b2qn−1 − bq
n−1
2
(
zq
1−n
2
+j + z−1q
n−1
2
−j))
=
n−1∏
j=0
(
1− bzqj
)(
1− bqn−1−j/z
)
= (bz, b/z; q)n.
We take similar limits on the right-hand side of equation (3.7).
Our final result concerns another cubic theta extension of Gessel and Stanton’s quadratic
summation [10, equation (1.4)].
Corollary 3.7. We have the following summation〈〈
az, a/z; q, p
1
3
〉〉
n
(cz, c/z; q, p)n
(
cq−
1
4 , cq
1
4 ; q, p
)
n
=
n∑
k=0
ckq
k
4
(k−2)+nk θ
(
cq
3
2
k− 3
4 ; p
)
θ
(
cq
k
2
− 3
4 ; p
) (q−n; q, p)k
(q; q, p)k
(
cq−
1
4 ; q
1
2 , p
)
k(
cqn−
1
4 ; q
1
2 , p
)
k
(
q
1
4 z, q
1
4 /z; q
1
2 , p
)
k
(cz, c/z; q, p)k
×
〈〈
acqn−1, aq1−k/c; q, p
1
3
〉〉
k
〈〈
aq
k
2
+ 1
4 , aq
k
2
− 1
4 ; q, p
1
3
〉〉
n−k.
Proof. We apply Theorem 2.15 to
f(z) =
〈〈
az, a/z; q, p
1
3
〉〉
n
(cz, c/z; q, p)n
. �
Acknowledgements
The work in this paper has been supported by FWF Austrian Science Fund grant F50-08 within
the SFB “Algorithmic and enumerative combinatorics”.
References
[1] Askey R., Wilson J., Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials,
Mem. Amer. Math. Soc. 54 (1985), no. 319, iv+55 pages.
[2] Baxter R.J., Eight-vertex model in lattice statistics and one-dimensional anisotropic Heisenberg chain.
II. Equivalence to a generalized ice-type model, Ann. Physics 76 (1973), 25–47.
[3] Bhargava S., Unification of the cubic analogues of the Jacobian theta-function, J. Math. Anal. Appl. 193
(1995), 543–558.
http://dx.doi.org/10.1090/memo/0319
http://dx.doi.org/10.1016/0003-4916(73)90440-5
http://dx.doi.org/10.1006/jmaa.1995.1252
Elliptic Hypergeometric Summations by Taylor Series Expansion and Interpolation 21
[4] Bhargava S., Fathima S.N., Laurent coefficients for cubic theta functions, South East Asian J. Math. Math.
Sci. 1 (2003), 27–31.
[5] Cooper S., The Askey–Wilson operator and the 6Φ5 summation formula, South East Asian J. Math. Math.
Sci. 1 (2002), 71–82.
[6] Cooper S., Toh P.C., Determinant identities for theta functions, J. Math. Anal. Appl. 347 (2008), 1–7.
[7] Date E., Jimbo M., Kuniba A., Miwa T., Okado M., Exactly solvable SOS models: local height probabilities
and theta function identities, Nuclear Phys. B 290 (1987), 231–273.
[8] Frenkel I.B., Turaev V.G., Elliptic solutions of the Yang–Baxter equation and modular hypergeometric
functions, in The Arnold–Gelfand Mathematical Seminars, Editors V.I. Arnold, I.M. Gelfand, V.S. Retakh,
M. Smirnov, Birkhäuser Boston, Boston, MA, 1997, 171–204.
[9] Gasper G., Rahman M., Basic hypergeometric series, Encyclopedia of Mathematics and its Applications,
Vol. 96, 2nd ed., Cambridge University Press, Cambridge, 2004.
[10] Gessel I., Stanton D., Applications of q-Lagrange inversion to basic hypergeometric series, Trans. Amer.
Math. Soc. 277 (1983), 173–201.
[11] Ismail M.E.H., The Askey–Wilson operator and summation theorems, in Mathematical Analysis, Wavelets,
and Signal Processing (Cairo, 1994), Contemp. Math., Vol. 190, Amer. Math. Soc., Providence, RI, 1995,
171–178.
[12] Ismail M.E.H., Rains E.M., Stanton D., Orthogonality of very-well-poised series, Int. Math. Res. Not., to
appear.
[13] Ismail M.E.H., Stanton D., Applications of q-Taylor theorems, J. Comput. Appl. Math. 153 (2003), 259–272.
[14] Ismail M.E.H., Stanton D., Some combinatorial and analytical identities, Ann. Comb. 16 (2012), 755–771.
[15] Marco J.M., Parcet J., A new approach to the theory of classical hypergeometric polynomials, Trans. Amer.
Math. Soc. 358 (2006), 183–214, math.CO/0312247.
[16] Rains E.M., BCn-symmetric Abelian functions, Duke Math. J. 135 (2006), 99–180, math.CO/0402113.
[17] Rains E.M., Transformations of elliptic hypergeometric integrals, Ann. of Math. 171 (2010), 169–243,
math.QA/0309252.
[18] Rosengren H., Sklyanin invariant integration, Int. Math. Res. Not. 2004 (2004), 3207–3232,
math.QA/0405072.
[19] Rosengren H., An elementary approach to 6j-symbols (classical, quantum, rational, trigonometric, and
elliptic), Ramanujan J. 13 (2007), 131–166, math.CA/0312310.
[20] Rosengren H., Schlosser M., Elliptic determinant evaluations and the Macdonald identities for affine root
systems, Compos. Math. 142 (2006), 937–961, math.CA/0505213.
[21] Schlosser M.J., A Taylor expansion theorem for an elliptic extension of the Askey–Wilson operator, in
Special Functions and Orthogonal Polynomials, Contemp. Math., Vol. 471, Amer. Math. Soc., Providence,
RI, 2008, 175–186, arXiv:0803.2329.
[22] Schultz D., Cubic theta functions, Adv. Math. 248 (2013), 618–697.
[23] Spiridonov V.P., Theta hypergeometric series, in Asymptotic Combinatorics with Application to Mathemat-
ical Physics (St. Petersburg, 2001), NATO Sci. Ser. II Math. Phys. Chem., Vol. 77, Editors V.A. Malyshev,
A.M. Vershik, Kluwer Acad. Publ., Dordrecht, 2002, 307–327, math.CA/0303204.
[24] Warnaar S.O., Summation and transformation formulas for elliptic hypergeometric series, Constr. Approx.
18 (2002), 479–502, math.QA/0001006.
[25] Weber H., Elliptische Functionen und Algebraische Zahlen, Vieweg u. Sohn, Braunschweig, 1891.
[26] Whittaker E.T., Watson G.N., A course of modern analysis, Cambridge Mathematical Library , Cambridge
University Press, Cambridge, 1996.
http://dx.doi.org/10.1016/j.jmaa.2008.05.054
http://dx.doi.org/10.1016/0550-3213(87)90187-8
http://dx.doi.org/10.1007/978-1-4612-4122-5_9
http://dx.doi.org/10.1017/CBO9780511526251
http://dx.doi.org/10.2307/1999351
http://dx.doi.org/10.2307/1999351
http://dx.doi.org/10.1090/conm/190/02300
http://dx.doi.org/10.1016/S0377-0427(02)00644-1
http://dx.doi.org/10.1007/s00026-012-0158-1
http://dx.doi.org/10.1090/S0002-9947-04-03620-7
http://dx.doi.org/10.1090/S0002-9947-04-03620-7
http://arxiv.org/abs/math.CO/0312247
http://dx.doi.org/10.1215/S0012-7094-06-13513-5
http://arxiv.org/abs/math.CO/0402113
http://dx.doi.org/10.4007/annals.2010.171.169
http://arxiv.org/abs/math.QA/0309252
http://dx.doi.org/10.1155/S1073792804141731
http://arxiv.org/abs/math.QA/0405072
http://dx.doi.org/10.1007/s11139-006-0245-1
http://arxiv.org/abs/math.CA/0312310
http://dx.doi.org/10.1112/S0010437X0600203X
http://arxiv.org/abs/math.CA/0505213
http://dx.doi.org/10.1090/conm/471/09213
http://arxiv.org/abs/0803.2329
http://dx.doi.org/10.1016/j.aim.2013.08.021
http://arxiv.org/abs/math.CA/0303204
http://dx.doi.org/10.1007/s00365-002-0501-6
http://arxiv.org/abs/math.QA/0001006
http://dx.doi.org/10.1017/CBO9780511608759
1 Introduction
1.1 Elliptic hypergeometric series
1.2 The Askey–Wilson operator
1.3 The well-poised and elliptic Askey–Wilson operator
2 Elliptic Taylor expansions and interpolation
2.1 A quadratic elliptic Taylor expansion theorem
3 Expansions involving cubic theta functions
References
|
| id | nasplib_isofts_kiev_ua-123456789-147739 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T18:34:12Z |
| publishDate | 2016 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Schlosser, M.J. Yoo, M. 2019-02-15T18:58:25Z 2019-02-15T18:58:25Z 2016 Elliptic Hypergeometric Summations by Taylor Series Expansion and Interpolation / M.J. Schlosser, M. Yoo // Symmetry, Integrability and Geometry: Methods and Applications. — 2016. — Т. 12. — Бібліогр.: 26 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 30E05; 33D15; 33D70; 33E05; 33E20 DOI:10.3842/SIGMA.2016.039 https://nasplib.isofts.kiev.ua/handle/123456789/147739 We use elliptic Taylor series expansions and interpolation to deduce a number of summations for elliptic hypergeometric series. We extend to the well-poised elliptic case results that in the q-case have previously been obtained by Cooper and by Ismail and Stanton. We also provide identities involving S. Bhargava's cubic theta functions. This paper is a contribution to the Special Issue on Orthogonal Polynomials, Special Functions and Applications.
 The full collection is available at http://www.emis.de/journals/SIGMA/OPSFA2015.html.
 nowledgements
 The work in this paper has been supported by FWF Austrian Science Fund grant F50-08 within
 the SFB “Algorithmic and enumerative combinatorics". en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Elliptic Hypergeometric Summations by Taylor Series Expansion and Interpolation Article published earlier |
| spellingShingle | Elliptic Hypergeometric Summations by Taylor Series Expansion and Interpolation Schlosser, M.J. Yoo, M. |
| title | Elliptic Hypergeometric Summations by Taylor Series Expansion and Interpolation |
| title_full | Elliptic Hypergeometric Summations by Taylor Series Expansion and Interpolation |
| title_fullStr | Elliptic Hypergeometric Summations by Taylor Series Expansion and Interpolation |
| title_full_unstemmed | Elliptic Hypergeometric Summations by Taylor Series Expansion and Interpolation |
| title_short | Elliptic Hypergeometric Summations by Taylor Series Expansion and Interpolation |
| title_sort | elliptic hypergeometric summations by taylor series expansion and interpolation |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/147739 |
| work_keys_str_mv | AT schlossermj elliptichypergeometricsummationsbytaylorseriesexpansionandinterpolation AT yoom elliptichypergeometricsummationsbytaylorseriesexpansionandinterpolation |