Elliptic Biorthogonal Polynomials Connected with Hermite's Continued Fraction
We study a family of the Laurent biorthogonal polynomials arising from the Hermite continued fraction for a ratio of two complete elliptic integrals. Recurrence coefficients, explicit expression and the weight function for these polynomials are obtained. We construct also a new explicit example of t...
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Vinet, L. Zhedanov, A. 2019-02-16T09:00:30Z 2019-02-16T09:00:30Z 2007 Elliptic Biorthogonal Polynomials Connected with Hermite's Continued Fraction / L. Vinet, A. Zhedanov // Symmetry, Integrability and Geometry: Methods and Applications. — 2007. — Т. 3. — Бібліогр.: 27 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 33C45; 42C05 https://nasplib.isofts.kiev.ua/handle/123456789/147832 We study a family of the Laurent biorthogonal polynomials arising from the Hermite continued fraction for a ratio of two complete elliptic integrals. Recurrence coefficients, explicit expression and the weight function for these polynomials are obtained. We construct also a new explicit example of the Szegö polynomials orthogonal on the unit circle. Relations with associated Legendre polynomials are considered. This paper is a contribution to the Vadim Kuznetsov Memorial Issue “Integrable Systems and Related Topics”. The authors thank to the referees for their remarks leading to improvement of the text. A.Zh. thanks Centre de Recherches Math´ematiques of the Universit´e de Montr´eal for hospitality. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Elliptic Biorthogonal Polynomials Connected with Hermite's Continued Fraction Article published earlier |
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Elliptic Biorthogonal Polynomials Connected with Hermite's Continued Fraction |
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Elliptic Biorthogonal Polynomials Connected with Hermite's Continued Fraction Vinet, L. Zhedanov, A. |
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Elliptic Biorthogonal Polynomials Connected with Hermite's Continued Fraction |
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Elliptic Biorthogonal Polynomials Connected with Hermite's Continued Fraction |
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Elliptic Biorthogonal Polynomials Connected with Hermite's Continued Fraction |
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Elliptic Biorthogonal Polynomials Connected with Hermite's Continued Fraction |
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elliptic biorthogonal polynomials connected with hermite's continued fraction |
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Vinet, L. Zhedanov, A. |
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Vinet, L. Zhedanov, A. |
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Symmetry, Integrability and Geometry: Methods and Applications |
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We study a family of the Laurent biorthogonal polynomials arising from the Hermite continued fraction for a ratio of two complete elliptic integrals. Recurrence coefficients, explicit expression and the weight function for these polynomials are obtained. We construct also a new explicit example of the Szegö polynomials orthogonal on the unit circle. Relations with associated Legendre polynomials are considered.
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Elliptic Biorthogonal Polynomials Connected with Hermite's Continued Fraction / L. Vinet, A. Zhedanov // Symmetry, Integrability and Geometry: Methods and Applications. — 2007. — Т. 3. — Бібліогр.: 27 назв. — англ. |
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AT vinetl ellipticbiorthogonalpolynomialsconnectedwithhermitescontinuedfraction AT zhedanova ellipticbiorthogonalpolynomialsconnectedwithhermitescontinuedfraction |
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2025-11-24T04:39:12Z |
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2025-11-24T04:39:12Z |
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1850842233168723968 |
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 3 (2007), 003, 18 pages
Elliptic Biorthogonal Polynomials Connected
with Hermite’s Continued Fraction?
Luc VINET † and Alexei ZHEDANOV ‡
† Université de Montréal, PO Box 6128, Station Centre-ville, Montréal QC H3C 3J7, Canada
E-mail: luc.vinet@umontreal.ca
‡ Donetsk Institute for Physics and Technology, Donetsk 83114, Ukraine
E-mail: zhedanov@yahoo.com
Received October 07, 2006, in final form December 12, 2006; Published online January 04, 2007
Original article is available at http://www.emis.de/journals/SIGMA/2007/003/
Abstract. We study a family of the Laurent biorthogonal polynomials arising from the Her-
mite continued fraction for a ratio of two complete elliptic integrals. Recurrence coefficients,
explicit expression and the weight function for these polynomials are obtained. We construct
also a new explicit example of the Szegö polynomials orthogonal on the unit circle. Relations
with associated Legendre polynomials are considered.
Key words: Laurent biorthogonal polynomials; associated Legendre polynomials; elliptic
integrals
2000 Mathematics Subject Classification: 33C45; 42C05
To the memory of Vadim B. Kuznetsov
Vadim Kuznetsov spent a number of years at the Centre de Recherches Mathématiques
of the Université de Montréal in the mid-90s. This is when we had the privilege
to have him as a colleague and the chance to appreciate his scientif ic and human
qualities. His smile, great spirit, penetrating insights and outstanding scientif ic con-
tributions will always be with us.
1 Introduction
We start with well known identity (see e.g. [24])
d2 snn u
du2
= n(n− 1) snn−2 u− n2(1 + k2) snn u+ n(n+ 1)k2 snn+2 u (1.1)
for the Jacobi elliptic function snu depending on an argument u and a modulus k. This (and
similar 11 formulas obtained if one replaces sn with other elliptic functions) identity goes back
to Jacobi and usually is exploited in order to establish recurrence relations for elliptic integrals.
Indeed, introduce the following elliptic integrals
Jn(x; k) =
∫ v
0
k2n sn2n u du =
∫ x
0
k2nt2ndt√
(1− t2)(1− k2t2)
, (1.2)
where snu = t and x = sn v. Then from (1.1) we obtain
(2n− 1)Jn(x; k)− (2n− 2)(k2 + 1)Jn−1(x; k) + k2(2n− 3)Jn−2(x; k)
?This paper is a contribution to the Vadim Kuznetsov Memorial Issue “Integrable Systems and Related Topics”.
The full collection is available at http://www.emis.de/journals/SIGMA/kuznetsov.html
mailto:luc.vinet@umontreal.ca
mailto:zhedanov@yahoo.com
http://www.emis.de/journals/SIGMA/2007/003/
http://www.emis.de/journals/SIGMA/kuznetsov.html
2 L. Vinet and A. Zhedanov
= k2n−2x2n−3
√
(1− x2)(1− k2x2). (1.3)
From this formula we can express Jn+1(x; k) in the form [12]
Jn+1(x; k) = Q(x;n)
√
(1− x2)(1− k2x2) +AnJ1(x; k)−BnJ0(x; k), (1.4)
where Q(x;n) is a polynomial in x (depending on n) and the coefficients An, Bn satisfy the
same recurrence relations as Jn+1(1; k), i.e.
(2n+ 1)An − 2n(1 + k2)An−1 + (2n− 1)k2An−2 = 0,
(2n+ 1)Bn − 2n(1 + k2)Bn−1 + (2n− 1)k2Bn−2 = 0 (1.5)
with obvious initial conditions
A−1 = 0, B−1 = −1, A0 = 1, B0 = 0. (1.6)
Note that from recurrence relations (1.5) and initial conditions (1.6) it follows that An(k) is
a polynomial in k2 of degree n and Bn(k) is a polynomial in k2 of degree n but having common
factor k2, i.e. Bn = k2Vn−1(k2), where Vn−1(z) is a polynomial of degree n − 1 in z for any
n = 1, 2, . . . .
There is an elementary Wronskian-type identity following directly from (1.5) [12]:
BnAn−1 −AnBn−1 =
k2n
2n+ 1
. (1.7)
Formula (1.4) allows to reduce calculation of any (incomplete) elliptic integrals of the form∫ x
0
P (t) dt√
(1− t2)(1− k2t2)
(where P (t) is a polynomial) to standard elliptic integrals of the first and second kind J0(x; k)
and J1(x; k). This result is well known since Jacobi.
In what follows we will denote J0(1; k) = K(k) and J1(1; k) = J(k). Note that K(k) is the
standard complete elliptic integral of the first kind [24] and
J(k) = K(k)− E(k) = −kdE
dk
,
where
E(k) =
∫ 1
0
(
1− k2x2
1− x2
)1/2
dx
is the complete elliptic integral of the second kind.
We note also an important relation with hypergeometric functions [24]:
K(k) =
π
2 2F1(1/2, 1/2; 1; k2), E(k) =
π
2 2F1(−1/2, 1/2; 1; k2),
J(k) =
πk2
4 2F1(3/2, 1/2; 2; k2). (1.8)
Hermite in his famous “Cours d’analyse” [12] (see also [10, 11]) derived a continued fraction
connected with a ratio of two complete elliptic integrals:
J(k)
K(k)
=
k2
2(1 + k2)−
9k2
4(1 + k2)−
25k2
6(1 + k2)− · · ·
, (1.9)
Elliptic Biorthogonal Polynomials Connected with Hermite’s Continued Fraction 3
where
K(k) =
∫ 1
0
dx√
(1− x2)(1− k2x2)
, J(k) =
∫ 1
0
k2x2 dx√
(1− x2)(1− k2x2)
.
The Hermite continued fraction (1.9) follows directly from (1.3). Note that the continued frac-
tion (1.9) belongs to the class of the so-called T-continued fractions (with respect to the variab-
le k2) introduced and studied by Thron (see [14] for details). Thus perhaps Hermite was the
first to introduce an explicit example of the T -continued fraction. In what follows we will see
that this example gives rise to a class of polynomials which are biorthogonal on the unit circle.
Rewrite relation (1.4) for x = 1 in the form
J(k)/K(k)−Bn/An =
Jn+1(k)
K(k)An
(1.10)
from which Hermite concluded that the rational function Bn/An is an approximate expression
for the ratio J(k)/K(k) to within terms of degree n+ 1 in k2.
Hermite also noted that the coefficientsAn, Bn appeared as power coefficients in the following
Taylor expansions:
J0(x)√
(1− x2)(1− k2x2)
= A0x+A1x
3 + · · ·+Anx
2n+1 + · · · ,
J1(x)√
(1− x2)(1− k2x2)
= B0x+B1x
3 + · · ·+Bnx
2n+1 + · · · . (1.11)
Thus formulas (1.11) can be considered as generating functions for An, Bn.
2 Laurent biorthogonal polynomials
Introduce the new variable z = k2 and define the polynomials Pn(z) = An/ξn of degree n in z,
where
ξn
ξn+1
=
2n+ 3
2n+ 2
, ξ0 = 1.
We have
ξn =
n!
(3/2)n
, (2.1)
where (a)n = a(a+ 1) · · · (a+ n− 1) is the standard Pochhammer symbol (shifted factorial).
Then it is seen that Pn(z) = zn + O(zn−1), i.e. Pn(z) are monic polynomials. Moreover
from (1.5) it follows that Pn(z) satisfy the 3-term recurrence relation
Pn+1(z) + dnPn(z) = z(Pn(z) + bnPn−1(z)), (2.2)
where
dn = −1, n = 0, 1, . . . , bn = −(n+ 1/2)2
n(n+ 1)
, n = 1, 2, . . . (2.3)
with initial conditions
P−1(z) = 0, P0(z) = 1. (2.4)
4 L. Vinet and A. Zhedanov
Define also the polynomials
P
(1)
n−1(z) = 2Bn/(zξn).
It is then easily verified that the polynomials P (1)
n (z) are again n-th degree monic polynomials
in z satisfying the recurrence relation
P
(1)
n+1(z) + dn+1P
(1)
n (z) = z(P (1)
n (z) + bn+1P
(1)
n−1(z)). (2.5)
The polynomials Pn(z) are Laurent biorthogonal polynomials (LBP) [7] and the polynomi-
als P (1)
n (z) are the corresponding associated LBP. The recurrence coefficients bn, dn completely
characterize LBP. The nondegeneracy condition bndn 6= 0, n = 1, 2, . . . [7, 27] obviously holds
in our case.
It is well known that LBP possess the biorthogonality property [7]. This means that there
exists a family of other LBP P̂n(z) and a linear functional σ such that
〈σ, Pn(z)P̂m(1/z)〉 = hmδnm, (2.6)
where the normalization constants hn are expressed as [7]
hn =
n∏
k=1
bk
dk
.
The linear functional σ is defined on the space of all monomials zs with both positive and
negative values of s:
cs = 〈σ, zs〉, s = 0,±1,±2, . . . , (2.7)
where cs is a sequence of moments (this sequence is infinite in both directions).
The biorthogonality condition (2.6) is equivalent (under the nondegeneracy condition hn 6= 0)
to the orthogonality relations
〈σ, Pn(z)z−j〉 = 0, j = 0, 1, . . . , n− 1. (2.8)
Note that in our case from (2.2) and (2.3) it follows that
Pn(0) = 1, n = 0, 1, 2, . . . . (2.9)
Introduce the reciprocal LBP by the formula [23]
P ∗
n(z) =
znPn(1/z)
Pn(0)
. (2.10)
It appears that P ∗
n(z) are again LBP with the recurrence coefficients [23]
b∗n =
bn
dndn+1
, d∗n =
1
dn
. (2.11)
The moments c∗n of the reciprocal polynomials are expressed as [23]
c∗n =
c1−n
c1
. (2.12)
In our case (2.3) we have b∗n = bn, d∗n = dn. Hence the reciprocal polynomials coincide with
the initial ones:
znPn(1/z) = Pn(z). (2.13)
Moreover in our case c1 = d0 = −1 and thus c∗n = −c1−n.
Elliptic Biorthogonal Polynomials Connected with Hermite’s Continued Fraction 5
The polynomials P̂n(z) are the biorthogonal partners with respect to the polynomials Pn(z).
Their moment sequence ĉn is obtained from the initial moment sequence by reflection:
ĉn = c−n, n = 0,±1,±2, . . . . (2.14)
Note that under the assumption c0 = 1 (standard normalization condition) we have ĉ0 = 1 as
well.
For the biorthogonal partners there is an explicit expression [7, 23]
P̂n(z) =
znPn+1(1/z)− zn−1Pn(1/z)
Pn+1(0)
. (2.15)
In our case (taking into account properties (2.9) and (2.13)) it is seen that
P̂n(z) =
Pn+1(z)− Pn(z)
z
. (2.16)
Formula (2.16) admits another interpretation if one introduces the Christoffel transform (CT)
of LBP. Recall [27, 23, 22] that the CT for LBP is defined as
P (C)
n (z) =
Pn+1(z)− UnPn(z)
z − µ
, n = 0, 1, . . . , Un =
Pn+1(µ)
Pn(µ)
, (2.17)
where µ is an arbitrary parameter such that Pn(µ) 6= 0, n = 1, 2, . . . . The polynomials P (C)
n (z)
are again monic LBP with the transformed recurrence coefficients
b(C)
n = bn
bn+1 + Un
bn + Un−1
, d(C)
n = dn
dn+1 + Un+1
dn + Un
. (2.18)
The moments c(C)
n corresponding to CT are expressed as [27]
c(C)
n =
cn+1 − µcn
c1 − µ
. (2.19)
There is a special case of the CT when µ = 0. In this case, we have
P (C)
n (z) =
Pn+1(z) + dnPn(z)
z
. (2.20)
For the recurrence coefficients in this case we have [27]
b̃n = bn
bn+1 − dn
bn − dn−1
, n = 1, 2, . . . ,
d̃0 = d0 − b1, d̃n = dn−1
bn+1 − dn
bn − dn−1
, n = 1, 2, . . . . (2.21)
Note that there is some “irregularity” in the expression for d̃n in (2.21) for n = 0 and n = 1, 2, . . . .
This irregularity can be avoided if one formally puts b0 = 0. However in our case b0 6= 0. Hence
we will indeed have such an irregularity in the analytic dependence of the coefficients d̃n in n.
Namely, we have the explicit expressions
b̃n = − (n+ 1/2)2
(n+ 1)(n+ 2)
, n = 1, 2, . . . ,
d̃0 = 1/8, d̃n = − n
n+ 2
, n = 1, 2, . . . . (2.22)
6 L. Vinet and A. Zhedanov
The corresponding moments are transformed simply as a “shift”
c(C)
n =
cn+1
c1
. (2.23)
Comparing (2.20) with (2.16) we see that in our case the biorthogonal partners P̂n(z) coincide
with the CT LBP P
(C)
n for µ = 0. Taking into account that in our case c1 = −1 we obtain that
the “negative” moments are expressed as
c−n = −cn+1, n = 1, 2, . . . . (2.24)
The LBP are connected with the two-point Padé approximation problem [7]. Given the
moments cn, n = 0,±1,±2, . . . , consider two formal power series
F+(z) =
∞∑
k=1
ckz
−k, F−(z) =
∞∑
k=0
c−kz
k. (2.25)
Then we have [7, 27]
P
(1)
n−1(z)
Pn(z)
=
F+(z)
c1
+O(z−n−1),
P
(1)
n−1(z)
Pn(z)
= −F−(z)
c1
+O(zn). (2.26)
It is convenient to introduce the formal Laurent series
F (z) =
F+(z)− F−(z)
c1
. (2.27)
Then we have
F (z)−
P
(1)
n−1(z)
Pn(z)
=
{
O(z−n−1), z →∞,
O(zn), z → 0,
(2.28)
i.e. LBP Pn(z), P (1)
n−1(z) solve the problem of two-point Padé approximation (at z = 0,∞). Note
that the Hermite formula (1.10) describes “one half” of this two-point Padé approximation
problem.
So it is reasonable to refer to the polynomials Pn(z) as the Hermite elliptic Laurent biorthog-
onal polynomials.
3 The weight function and biorthogonality
Consider the functions
F−(z) = − 2J(k)
k2K(k)
, F+(z) = −2J(1/k)
K(1/k)
, (3.1)
where z = k2. It is assumed that the function F−(z) is defined near z = 0 while the function
F+(z) is defined near z = ∞. Note that formally
F+(z) =
F−(1/z)
z
.
From the considerations of the previous section it is easily verified that formula (2.28) holds
for the LBP defined by the recurrence coefficients (2.3). It is seen also that if one introduces
the function (cf. [9])
w(z) =
F+(z)− F−(z)
2πiz
(3.2)
Elliptic Biorthogonal Polynomials Connected with Hermite’s Continued Fraction 7
then ∫
C
z−kw(z)dz = ck, k = 0,±1,±2, . . . , (3.3)
where the integration contour C is the unit circle. Thus the biorthogonality property (2.6) can
be presented in the form∫
C
Pn(z)P̂m(1/z)w(z)dz = hnmδnm. (3.4)
Now we calculate the weight function w(z) in a more explicit form. We have
w(z) =
1
πik2
(
− J(k)
k2K(k)
+
J(1/k)
K(1/k)
)
. (3.5)
Using the relation J(k) = K(k)−E(k) and the formulas for the complete elliptic integrals with
inverse modulus (as usual K ′(k) = K(k′), E′(k) = E(k′), k′2 = 1− k2):
K(1/k) = k(K(k) + iK ′(k)), E(1/k) =
E(k)− iE′(k)− k′2K(k) + ik2K ′(k)
k
we arrive at the formula
w(z) =
−K(k)K ′(k) + E(k)K ′(k) +K(k)E′(k)
k4K(k)K(1/k)
. (3.6)
The latter expression can be further simplified using the Legendre relation [24]
−K(k)K ′(k) + E(k)K ′(k) +K(k)E′(k) = π/2
to give
w(z) =
1
2z3/2
1
K(k)K(1/k)
. (3.7)
Now we introduce variable θ on the unit circle such that k = z1/2 = eiθ/2. The biorthogonality
relation can then be written as∫ 2π
0
Pn(eiθ)P̂m(e−iθ)ρ(θ)dθ = hnδnm, (3.8)
where the weight function is
ρ(θ) =
i
2eiθ/2|K(eiθ/2)|2
. (3.9)
4 Generating function and explicit expression
From (1.11) we obtain the generating function for the corresponding LBP
Φ(x, z) =
F (x; z)√
(1− x2)(1− zx2)
=
∞∑
n=0
ξnPn(z), (4.1)
where ξn is given by (2.1) and
F (x; z) =
∫ x
0
dt√
(1− t2)(1− zt2)
is the standard (incomplete) elliptic integral of the first kind.
8 L. Vinet and A. Zhedanov
In order to find the explicit expression for the polynomials Pn(z) from (4.1), we note that if
1√
(1− x2)(1− zx2)
=
∞∑
n=0
βnx
2n (4.2)
then, obviously,
F (x; z) =
∞∑
n=0
βnx
2n+1/(2n+ 1)
and hence Φ(x, z) =
∞∑
n=0
An(z)x2n+1 where
An(z) =
n∑
s=0
βsβn−s
2s+ 1
. (4.3)
Formula (4.3) gives an explicit expression for the polynomials An(z) and hence for the LBP
Pn(z) if the coefficients βn are known. But it is easy to verify (using e.g. the binomial theorem)
that
βn =
(1/2)n
n! 2F1(−n, 1/2; 1/2− n; z). (4.4)
We thus have
An(z) =
n∑
m=0
(1/2)n(1/2)m(−n)m
(2m+ 1)n!m!(1/2− n)m
× 2F1(−m, 1/2; 1/2−m; z)2F1(m− n, 1/2; 1/2 +m− n; z). (4.5)
Another explicit expression is obtained if one notices that
1√
(1− x2)(1− k2x2)
=
∞∑
n=0
knx2nYn
(
k + k−1
2
)
, (4.6)
where Yn(z) are the ordinary Legendre polynomials [15]:
Yn(t) = 2F1
(
−n, n+ 1; 1;
1− t
2
)
.
We thus have the rather simple expression
An(z) = kn
n∑
s=0
Ys(q)Yn−s(q)
2s+ 1
, (4.7)
where q = (k + k−1)/2 (recall that k2 = z).
For z = 1 we have βn = 1 for all n in (4.2). Hence we have from (4.3)
An(1) = Gn, (4.8)
where we denote
Gn =
n∑
s=0
1
2s+ 1
Elliptic Biorthogonal Polynomials Connected with Hermite’s Continued Fraction 9
– the finite sum of inverse odd numbers. Gn can be obviously expressed in terms of the Euler
“harmonic numbers” defined as
Hn =
n∑
k=1
1/k.
We have clearly
Gn = H2n+1 −Hn/2. (4.9)
Consider the recurrence relation of the type (1.5)
(2n+ 1)ψn − 2n(1 + z)ψn−1 + (2n− 1)zψn−2 = 0. (4.10)
For z = 1 we see that ψn = Gn is a solution of this equation. The second independent solution
for z = 1 is trivial – it is a constant: ψn = const. This means that the general solution of the
equation (4.10) for z = 1 can be presented in the form
ψn = α+ βGn (4.11)
with arbitrary constants α, β. In particular, for Bn(1) we can write
Bn(1) = Gn − 1. (4.12)
5 Polynomials orthogonal on the unit circle
Consider the Christoffel transform (2.17) of our polynomials with µ = 1. We have
Un =
Pn+1(1)
Pn(1)
=
ξnAn+1(1)
ξn+1An(1)
=
(n+ 3/2)Gn+1
(n+ 1)Gn
. (5.1)
For the corresponding transformed LBP, we have the expression
P̃n(z) =
Pn+1(z)− UnPn(z)
z − 1
. (5.2)
The moments are calculated by (2.19):
c̃n =
cn+1 − cn
c1 − 1
=
cn − cn+1
2
. (5.3)
Using property (2.24) we see that c̃−n = c̃n, i.e. the moments c̃n are symmetric with respect to
reflection. In turn, this is equivalent to the statement that the corresponding polynomials P̃n(z)
are the Szegö polynomials which are orthogonal on the unit circle [21, 5]:∫ 2π
0
P̃n(eiθ)P̃m(e−iθ)ρ̃(θ)dθ = hnδnm, (5.4)
where
ρ̃(θ) =
ρ(θ)
c1 − 1
(eiθ − 1). (5.5)
In our case we have explicitly (see (3.9))
ρ̃(θ) =
sin(θ/2)
2|K(eiθ/2)|2
. (5.6)
10 L. Vinet and A. Zhedanov
It is well known that polynomials orthogonal on the unit circle are defined by the recurrence
relation [5]
Pn+1(z) = zPn(z)− anP
∗
n(z), n = 0, 1, . . . , (5.7)
where P ∗
n(z) = znP̄n(1/z) (bar means complex conjugate). In our case all moments c̃n are real,
hence P̃ ∗
n(z) = znP̃n(1/z). The parameters an are called the reflection parameters. They play
a crucial role in the theory of Szegö polynomials on the unit circle. We have
an = −P̃n+1(0). (5.8)
It is well known [5] that if the reflection parameters are real and satisfy the condition |an| < 1
for all n = 0, 1, . . . then the positive weight function ρ(θ) > 0 always exists. Moreover in this
case the weight function is symmetric on the unit circle: ρ(2π − θ) = ρ(θ).
In our case we have
an = −P̃n+1(0) = −Un+1Pn+1(0) + Pn+2(0) = 1− Un+1, (5.9)
where we used the property Pn(0) = 1.
It is easily seen that
an = − 1
2(n+ 2)
− 1
2(n+ 2)G(n+ 1)
, n = 0, 1, . . . . (5.10)
In (5.10) both terms are negative and less then 1/2 in absolute value. Hence −1 < an < 0 for
all n = 0, 1, . . . and the function ρ̃(θ) is positive as is seen from (5.6).
We thus have a (presumably) new example of the Szegö polynomials orthogonal on the unit
circle for which both weight function and recurrence coefficients are known explicitly.
Following [4] and [26], to any polynomials P̃n(z) orthogonal on the unit circle with the
property −1 < an < 1 one can associate symmetric monic polynomials Sn(x) = xn + O(xn−1)
orthogonal on an interval of the real axis. Explicitly
Sn(x) =
z−n/2(P̃n(z) + P̃ ∗
n(z))
1− an−1
, (5.11)
where x = z1/2 + z−1/2 (it is assumed that one chooses one branch of the function z1/2 such
that for z = reiθ we have z1/2 = r1/2eiθ/2, −π < θ < π). The polynomials Sn(x) satisfy the
three-term recurrence relation
Sn+1 + unSn−1(x) = xSn(x), (5.12)
where the recurrence coefficients are
un = (1 + an−1)(1− an−2), n = 1, 2, . . . . (5.13)
In (5.13) it is assumed that a−1 = −1 (this is a standard convention in the theory of polynomials
orthogonal on the unit circle), so u1 = 2(1 + a0).
If the polynomials P̃n(z) are orthogonal on the unit circle∫ 2π
0
P̃n(eiθ)P̃ ∗
m(e−iθ)ρ(θ)dθ = 0, m 6= n (5.14)
with the weight function ρ(θ) then polynomials Sn(x) are orthogonal on the symmetric interval
[−2, 2]∫ 2
−2
Sn(x)Sm(x)w(x)dx = hnδnm (5.15)
Elliptic Biorthogonal Polynomials Connected with Hermite’s Continued Fraction 11
with the weight function [4, 26]
w(x) =
ρ(θ)
sin(θ/2)
, (5.16)
where x = 2 cos(θ/2).
In our case it is elementary verified that
un =
(n+ 1/2)2
n(n+ 1)
, n = 1, 2, . . . . (5.17)
These recurrence coefficients correspond to well known associated Legendre polynomials studied,
e.g. in [1]. Recall that generic associated Legendre polynomials are symmetric OP satisfying the
recurrence relation (5.12) with the recurrence coefficients
un =
(n+ ν)2
(n+ ν)2 − 1/4
(5.18)
with arbitrary nonnegative parameter ν. The ordinary Legendre polynomials correspond to
ν = 0. In our case we have ν = 1/2.
Consider the weight function for these polynomials. From (5.16) and (5.6) we derive
w(x) =
1
2|K(e−iθ/2)|2
. (5.19)
We can simplify this formula if one exploits the relations for K(z) where |z| = 1. Namely, one
has [17]
K(e±iφ) =
1
2
e∓iφ/2(K(cos(φ/2))± iK(sin(φ/2))), −π < φ ≤ π,
whence we have
w(x) =
2
K2(cos(θ/4)) +K2(sin(θ/4))
. (5.20)
Taking into account that x = 2 cos(θ/2) we finally arrive at the formula
w(x) =
2
K2(
√
1/2 + x/4) +K2(
√
1/2− x/4)
, −2 ≤ x ≤ 2. (5.21)
Note that the function w(x) is even w(−x) = w(x) as should be for symmetric polynomials. It
has the only maximum at x = 0: w(0) = 1/K2(
√
1/2) = 16π
Γ4(1/4)
. Near the endpoints of the
interval [−2, 2] the weight function w(x) tends to zero rapidly.
The weight function for generic associated Legendre polynomials (with arbitrary ν) was
found in [1]. Our formula (5.21) can be obtained from the results of [1] by putting ν = 1/2.
Note, that Pollaczek studied [18] more general orthogonal polynomials containing 4 parameters.
The associated Legendre polynomials (as well as the associated ultraspherical polynomials) are
contained in the Pollaczek polynomial family as a special case.
Similar polynomials were studied also in [19] where the author in fact rediscovered the Her-
mite approach (as well Hermite’s continued fraction (1.9)) to elliptic integrals. He consid-
ered integrals Jn(k) as moments for some “elliptic” orthogonal polynomials having w(x) =
1/
√
(1− x2)(1− k2x2) as an orthogonality weight on the interval [−1, 1]. As a by-product
the author of [19] introduced polynomials which are similar to the Hermite polynomials An(z),
12 L. Vinet and A. Zhedanov
Bn(z). He then related them with the associated Legendre polynomials in a similar way. It is in-
teresting to note that Hermite himself already introduced such polynomials in [10, 11]. Hermite
also established differential properties of these polynomials anticipating results of Rees [19].
Consider relations between LBP of special type and orthogonal polynomials on an interval
in details.
We have P̃n(z) = (Pn+1(z)−UnPn(z))/(z− 1). Substituting this formula to (5.11) and using
the inversion property znPn(1/z) = Pn(z) of our LBP Pn(z) we immediately obtain a very
simple relation
Sn(x) = z−n/2Pn(z). (5.22)
We can see this also using recurrence relation for the LBP
Pn+1(z) + dnPn(z) = z(Pn(z) + bnPn−1(z)). (5.23)
Assume (as in our case) that dn = −1, n = 0, 1, . . . . Then Pn(0) = 1 for all n = 0, 1, 2, . . . and
from (2.11) we obtain that the reciprocal polynomials coincide with the initial ones:
znPn(1/z) = Pn(z). (5.24)
Conversely, assume that some LBP are reciprocal (5.24). Then from (2.11) it follows that either
dn = 1, n = 0, 1, . . . or dn = −1, n = 0, 1, . . . . But we have obviously Pn(0) = 1 which leads to
the only possibility dn = −1, n = 0, 1, 2, . . . . Thus condition dn = −1 for all n is necessary and
sufficient for LBP to be reciprocal invariant (5.24). In this case polynomials
Sn(x) = z−n/2Pn(z) (5.25)
are obviously monic polynomials in x = z1/2 + z−1/2. From recurrence relation (5.23) with
dn = −1 we obtain recurrence relation for polynomials Sn(x)
Sn+1(x) + unSn−1(x) = xSn(x), (5.26)
where un = −bn. Thus we arrived at the same symmetric polynomials on the interval as in
the case of polynomials orthogonal on the unit circle. Note that for the LBP Pn(z) with the
property (5.24) we have c1 = d0 = −1 and c∗n = cn thus from (2.12) we obtain
c1−n = −cn (5.27)
for all n = 0,±1,±2, . . . . Perform now the Christoffel transform with µ = 1
P̃n(z) =
Pn+1(z)− UnPn(z)
z − 1
. (5.28)
From (2.19) and (5.27) we obtain that transformed moments are symmetric c̃−n = cn. This
means that polynomials P̃n(z) will satisfy the recurrence relation (5.7) with reflection param-
eters an given by an = 1 − Un+1 = 1 − Pn+2(1)/Pn+1(1). Formula (5.11) now is equivalent to
formula (5.25). Thus starting from arbitrary LBP with the property dn = −1 we can arrive at
the same symmetric OP on the interval Sn(x) as for the case of polynomials orthogonal on the
unit circle. From another point of view these relations are discussed also in [2].
Elliptic Biorthogonal Polynomials Connected with Hermite’s Continued Fraction 13
6 Geronimus transform. Laurent biorthogonal polynomials
with a concentrated mass added to the measure
In the previous section we showed that the Christoffel transformation of Hermite’s elliptic LBP
gives polynomials orthogonal on the unit circle with explicit reflection coefficients (5.10) and
the weight function given by (5.6). In this section we consider another spectral transformation
of Hermite’s elliptic LBP which is called the Geronimus transform (GT). This transform was
introduced in [27] and is similar to well known Geronimus transform for the ordinary orthogonal
polynomials [20, 25]. Recall thatGT for LBP is defined as [27]
P̃n(z) = VnPn(z) + z(1− Vn)Pn−1(z), (6.1)
where
V0 = 1, Vn =
µ
µ− φn/φn−1
, n = 1, 2, . . . . (6.2)
In (6.2) µ is an arbitrary parameter and φn is an arbitrary solution of the recurrence relation
φn+1 + dnφn = µ(φn + bnφn−1). (6.3)
Note that (6.3) is the same recurrence relation that (2.2) for LBP. Hence its the general solution
(up to a common factor) can be presented in the form
φn = Pn(µ) + χP
(1)
n−1(µ). (6.4)
It is easy to verify that the polynomials P̃n(z) are again LBP satisfying recurrence relation
P̃n+1(z) + d̃nP̃n(z) = z(P̃n(z) + b̃nP̃n−1(z)), (6.5)
where the recurrence coefficients are
b̃1 = χ(V1 − 1), b̃n = bn−1
1− Vn
1− Vn−1
, d̃n = dn
Vn+1
Vn
. (6.6)
It can be shown that CT and GT are reciprocal to one another [27]. This observation allows
one to obtain explicit biorthogonality relation for polynomials P̃n(z) starting from that for
polynomials Pn(z). Namely, the pair of the Stieltjes functions F+(z), F−(z) is transformed
as [27]
F̃+(z) =
νF+(z) + µ+ ν
z − µ
, F−(z) =
νF−(z)− µ− ν
z − µ
, (6.7)
where
ν =
µχ
c1 − χ
=
µχ
d0 − χ
.
Assume that |µ| ≤ 1. Choose the contour C as the unit circle (if |µ| = 1 we choose C as the
unit circle with a small deformation near z = µ in order to include the point z = µ inside the
contour C). Then the weight function for new polynomials P̃n(z) is defined as in (3.2), i.e.
w̃(z) =
F̃+(z)− F̃−(z)
2πiz
=
w(z)
z − µ
+
2(µ+ ν)
2πiz(z − µ)
. (6.8)
The second term in (6.8) will give a concentrated mass at the point z = µ added to a “regular”
part presented by the first term in (6.8).
14 L. Vinet and A. Zhedanov
Assume that µ = 1, as in our case. Then we have biorthogonality relation∫ 2π
0
ρ̃(θ)P̃n(eiθ) ˆ̃Pn(e−iθ)dθ +M P̃n(1) ˆ̃Pn(1) = 0, n 6= m, (6.9)
where
ρ̃(θ) =
ρ(θ)
eiθ − 1
(6.10)
is the “regular” of the weight function on the unit circle and the last term (6.10) describes the
concentrated mass
M =
ν + 1
2π
(6.11)
inserted at the point z = 1.
In our case we have dn = −1, n = 0, 1, . . . hence ν = −χ/(χ+ 1) and
M =
1
2π(1 + χ)
.
For the recurrence coefficients we have explicit formulas (6.6) where we need first to calculate
the coefficients Vn. We see that
φn = Pn(1) + χP
(1)
n−1(1).
But the values Pn(1) and P (1)
n−1(1) were already calculated (see (4.8), (4.12)). We thus have
φn =
(3/2)n
n!
(Gn + 2χ(Gn − 1)). (6.12)
Now all coefficients Vn, b̃n, d̃n are calculated explicitly.
We thus constructed a nontrivial example of the Laurent biorthogonal polynomials with
explicit both recurrence coefficients and the measure. The weight function for these polynomials
has a concentrated mass on the unit circle. Note that in contrast to the Christoffel transformed
Hermite’s polynomials the polynomials P̃n(z) constructed in this section are not polynomials of
the Szegö type. This means, in particular, that the biorthogonal partners ˆ̃Pn(z) do not coincide
with P̃n(z).
7 Associated families of the Laurent biorthogonal polynomials
Return to the sequence Jn(x, k) of incomplete elliptic integrals defined by (1.2). They satisfy
three-term recurrence relation (1.3). Repeating previous considerations we can express Jn(x; k)
for any n = 1, 2, . . . in terms of Jj(x; k) and Jj+1(x; k) for some fixed nonnegative integer j
(in (1.4) the case j = 0 is chosen):
Jn+1(x; k) = Q(x;n, j)
√
(1− x2)(1− k2x2) +A
(j)
n−jJj+1(x; k)−B
(j)
n−jJj(x; k) (7.1)
with some coefficients A(j)
n , B(j)
n .
We first note that there is an explicit expression of complete elliptic integrals Jn(k) in terms
of the Gauss hypergeometric function
Jn(k) =
k2nπ(1/2)n
2n! 2F1(1/2, 1/2 + n; 1 + n; k2) (7.2)
Elliptic Biorthogonal Polynomials Connected with Hermite’s Continued Fraction 15
(relation (7.2) can be easily verified by direct integration).
Repeating similar considerations that were already exploited in the two first sections we can
show that A(j)
n (z) and B(j)
n are determined by the recurrence relations
(n+ 3/2 + j)A(j)
n+1 = (n+ j + 1)(z + 1)A(j)
n − z(n+ j + 1/2)A(j)
n−1,
(n+ 3/2 + j)B(j)
n+1 = (n+ j + 1)(z + 1)B(j)
n − z(n+ j + 1/2)B(j)
n−1 (7.3)
with the same initial conditions as (1.6). Hence, A(j)
n (z) polynomials of n-th degree in z = k2
and B(j)
n (z) is a polynomial of degree n− 1 multiplied by z.
Introduce also the function
F (z; j) =
Jj+1(k)
Jj(k)
, (7.4)
where as usual we put k2 = z.
Then we will have the property
F (z; j)− B
(j)
n (z)
A
(j)
n (z)
= O(zn+1). (7.5)
In the same way it is possible to derive the generating functions for polynomials A(j)
n (z),
B
(j)
n (z):
(2j + 1)
k−2jJj(x; k)√
(1− x2)(1− k2x2)
=
∞∑
n=j
A
(j)
n−j(z)x
2n+1,
(2j + 1)
k−2jJj+1(x; k)√
(1− x2)(1− k2x2)
=
∞∑
n=j
B
(j)
n−j(z)x
2n+1. (7.6)
Using (4.6) we arrive at the explicit representation for polynomials A(j)
n (z) in terms of the
Legendre polynomials Yn(q):
A(j)
n (z) = (2j + 1)kn
n∑
s=0
Ys(q)Yn−s(q)
2s+ 2j + 1
, (7.7)
where z = k2, q = (k + 1/k)/2.
Now we introduce the monic LBP P
(j)
n (z) = A
(j)
n (z)/ξn where ξn = (j+1)n/(j+3/2)n. They
satisfy the recurrence relation of type (2.2) with
dn = −1, bn = − (n+ j + 1/2)2
(n+ j)(n+ j + 1)
. (7.8)
Thus we have j-associated polynomials with respect to the Hermite elliptic LBP, i.e. we should
replace n→ n+ j in formulas for recurrence coefficients.
Due to the property dn = −1 the associated polynomials P (j)
n again possess the invariance
property znP
(j)
n (1/z) = P
(j)
n (z). We thus can construct polynomials orthogonal on the unit
circle using formula (5.28). In order to get explicit expression for reflection parameters a(j)
n we
need the value P (j)
n (1). But this can easily be obtained from (7.7):
A(j)
n (1) = Gn(j) (7.9)
16 L. Vinet and A. Zhedanov
where we introduced the function
Gn(j) = (2j + 1)
n∑
s=0
1
2s+ 2j + 1
.
(Up to a common factor Gn(j) is a sum of n+ 1 succeeding inverse odd numbers starting from
1/(2j + 1); for j = 0 it coincides with Gn.) Thus
Un =
P
(j)
n+1(1)
P
(j)
n (1)
=
n+ 3/2 + j
n+ j + 1
Gn(j)
Gn+1(j)
and
a(j)
n = 1− Un+1.
Corresponding symmetric polynomial S(j)
n (x) on the interval satisfy recurrence relation (5.26)
with
un =
(n+ j + 1/2)2
(n+ j)(n+ j + 1)
(7.10)
so they coincide with the associated Legendre polynomials considered in [1]. Indeed, from
formula (5.18) we see that the “shift” parameter ν = j + 1/2 where j = 0, 1, 2, . . . .
It is interesting to note that recurrence relations for polynomials P (j)
n can be presented in
such a form that the all coefficients are linear in n – see e.g. (7.3). In [6] we considered a family
of such LBP connected with so-called generalized eigenvalue problem on su(1, 1) Lie algebra. In
our case, however, corresponding representations of su(1, 1) will not be unitary, in contrast to [6].
This leads to an interesting open problem how to describe associated classical LBP in terms of
non-unitary representations of su(1, 1) algebra. Note that generic orthogonal polynomials with
linear recurrence coefficients in n were studied in details by Pollaczek [18] who derived explicit
expression for them and found the weight function as well.
Note finally that LBP with recurrence coefficients (7.8) belong to a family of so-called “asso-
ciated Jacobi Laurent polynomials” introduced and studied by Hendriksen [8, 9]. Nevertheless,
in our case the parameters of the associated LBP belong to the exceptional class which was
not considered in [8, 9]. This means that in some formulas in [8, 9] the bottom parameter in
the Gauss hypergeometric function 2F1(z) takes negative integer values. In this case formulas
obtained by Hendriksen should be rederived in a different form. In particular, a simple explicit
expression of the power coefficients (in terms of the hypergeometric function 4F3(1)) for as-
sociated LBP obtained in [8] seems not to be valid in our case. Instead, we obtained explicit
expressions like (7.7).
8 Laurent biorthogonal polynomials connected
with the Stieltjes–Carlitz elliptic polynomials
Return to formula (1.1). If one denotes
rn(p) =
∫ ∞
0
k2nsn2n(t)e−ptdt (8.1)
then we obtain the recurrence relation
p2rn = 2n(2n+ 1)rn+1 − 4(1 + k2)n2rn + 2n(2n− 1)rn−1. (8.2)
Elliptic Biorthogonal Polynomials Connected with Hermite’s Continued Fraction 17
Again it is seen that for every n > 0 one can present
rn+1 = Anr1 −Bnr0, (8.3)
where obviously r0 =
∫∞
0 e−ptdt = p−1 and An, Bn satisfy the same recurrence relations as rn+1
e.g.
p2An = 2(n+ 1)(2n+ 1)k2An−1 − 4(1 + k2)(n+ 1)2An + 2(n+ 1)(2n+ 3)An+1 (8.4)
with initial conditions A0 = 1, A−1 = 0, B0 = 0, B−1 = −1. Now it is seen that An are
polynomials of degree n in both variables p2 and k2. If p = 0 then An become polynomials in k2
introduced by Hermite.
Coefficients An considered as polynomials in p2 become orthogonal polynomials because they
satisfy three-term recurrence relation typical for orthogonal polynomials. Orthogonal polynomi-
als of such type (and several related ones) where introduced and studied by Carlitz. He exploited
some explicit continued fractions found by Stieltjes. These continued fractions are connected
with elliptic functions (for details see, e.g. [13]). Today these orthogonal polynomials are known
by Stieltjes–Carlitz elliptic polynomials [16, 3, 13]. Note that the Stieltjes continued fraction is
obtained from (8.2) by the same way as Hermite obtained his continued fraction (1.9) for a ratio
of two elliptic integrals. Consider now polynomials An as LBP with respect to the argument
z = k2. Passing from An to monic polynomials Pn(z) (by the same way as for the case p = 0)
we arrive at the recurrence relation (2.2) with
dn = −1− p2
4(n+ 1)2
, bn = −(n+ 1/2)2
n(n+ 1)
. (8.5)
We see that the recurrence coefficient bn is the same as for the Hermite LBP, but the coefficient dn
now depends on n. This means that polynomials Pn(z) do not possess symmetric property
like (2.13).
In contrast to the case p = 0 the polynomials Pn(z) have more complicated properties. For
example, the reciprocal polynomials P ∗
n(z) defined by (2.10) do not belong to the same class,
their recurrence coefficients appear to be
d∗n = − 1
1 + p2
4(n+1)2
, b∗n = − (n+ 1)(n+ 2)2(n+ 1/2)2
((n+ 2)2 + p2/4)((n+ 1)2 + p2/4)
. (8.6)
The biorthogonal partners P̂n(z) defined by (2.15) have the recurrence coefficients
d̂n = − n(n+ 1)(p2(n+ 1)− n− 2)
(n(p2 − 1)− 1)((n+ 2)2 + p2/4)
,
b̂n = − (n+ 1/2)2(n+ 1)2(p2(n+ 1)− n− 2)
(n(p2 − 1)− 1)((n+ 1)2 + p2/4)((n+ 2)2 + p2/4)
. (8.7)
Hence biorthogonal partners P̂n(z) also do not belong to the same class that initial LBP Pn(z).
For the ordinary Stieltjes–Carlitz elliptic orthogonal polynomials the orthogonality measure
can be found explicitly: it is purely discrete one (see [3, 13]). The problem of finding the
orthogonality measure for the corresponding LBP seems to be much more complicated.
Acknowledgments
The authors thank to the referees for their remarks leading to improvement of the text. A.Zh.
thanks Centre de Recherches Mathématiques of the Université de Montréal for hospitality.
18 L. Vinet and A. Zhedanov
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147.
1 Introduction
2 Laurent biorthogonal polynomials
3 The weight function and biorthogonality
4 Generating function and explicit expression
5 Polynomials orthogonal on the unit circle
6 Geronimus transform. Laurent biorthogonal polynomials with a concentrated mass added to the measure
7 Associated families of the Laurent biorthogonal polynomials
8 Laurent biorthogonal polynomials connected with the Stieltjes-Carlitz elliptic polynomials
References
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