On transformation formulae for theta hypergeometric functions
Using an identity and certain summation formulas for truncated theta hypergeometric series, we establish transformation formulas for finite bilateral theta hypergeometric series.
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| Дата: | 2012 |
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| Мова: | Англійська |
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Institute of Mathematics, NAS of Ukraine
2012
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860508568808062976 |
|---|---|
| author | Denis, R. Y. Singh, S. N. Singh, S. P. Деніс, Р. Ю. Сінгх, С. Н. Сінгх, С. П. |
| author_facet | Denis, R. Y. Singh, S. N. Singh, S. P. Деніс, Р. Ю. Сінгх, С. Н. Сінгх, С. П. |
| author_sort | Denis, R. Y. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:31:34Z |
| description | Using an identity and certain summation formulas for truncated theta hypergeometric series, we establish transformation
formulas for finite bilateral theta hypergeometric series. |
| first_indexed | 2026-03-24T02:27:17Z |
| format | Article |
| fulltext |
UDC 517.5
R. Y. Denis (Univ. Gorakhpur, India),
S. N. Singh, S. P. Singh (T.D.P.G. College, Jaunpur, India)
ON TRANSFORMATION FORMULAE
FOR THETA HYPERGEOMETRIC FUNCTIONS
ПРО ФОРМУЛИ ПЕРЕТВОРЕННЯ
ДЛЯ ГIПЕРГЕОМЕТРИЧНИХ ТЕТА-ФУНКЦIЙ
Using an identity and certain summation formulas for truncated theta hypergeometric series, we establish transformation
formulas for finite bilateral theta hypergeometric series.
За допомогою однiєї тотожностi та формул пiдсумовування скорочених гiпергеометричних тета-рядiв встановлено
формули перетворення для скiнченних двостороннiх гiпергеометричних тета-рядiв.
1. Introduction, notations and definitions. Elliptic hypergeometric series and their extensions to
theta hypergeometric series has become an increasingly active area of research these days. In the
present paper, we have established transformation formulae for bilateral theta hypergeometric series.
Special cases of the results established in this paper have also been deduced.
A modified Jacobi’s theta function with argument x and nome p is defined by
θ(x; p) = [x; p]∞[p/x; p]∞ ≡ [x, p/x; p]∞. (1.1)
Also
θ(x1, x2, . . . , xr; p) = θ(x1; p)θ(x2; p) . . . θ(xr; p)
and
[x; p]∞ =
∞∏
r=0
(1− xpr).
Following Gasper and Rahman [1] (Chapter 11, (11.2.5) and (11.2.53)) theta shifted factorial is
defined by
[a; p, q]n =
θ(a; p)θ(aq; p) . . . θ(aq
n−1; p), n > 0,
1, n = 0.
Also
[a; q, p]−n =
qn(n+1)/2
(−a)n[q/a; q, p]n
, n ≥ 1, (1.2)
and
[a1, a2, . . . , ar; q, p]n = [a1; q, p]n[a2; q, p]n . . . [ar; q, p]n. (1.3)
Corresponding to Spiridonov [2], theta hypergeometric series is defined by
c© R. Y. DENIS, S. N. SINGH, S. P. SINGH, 2012
994 ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 7
ON TRANSFORMATION FORMULAE FOR THETA HYPERGEOMETRIC FUNCTIONS 995
r+1Er
[
a1, a2, . . . , ar+1; q, p; z
b1, b2, . . . , br
]
=
∞∑
n=0
[a1, a2, . . . , ar+1; q, p]nz
n
[q, b1, b2, . . . , br; q, p]n
, (1.4)
where max{|z|, |q|, |p|} < 1.
Corresponding to Spiridonov [3] a very well-poised theta hypergeometric series is defined by
r+1Vr[a1; a6, . . . , ar+1; q, p; z] =
=
∞∑
n=0
θ(a1q
2n; p)[a1, a6, . . . , ar+1; q, p; ]n
θ(a1; p)[a1q/a6, . . . , a1q/ar+1; q, p]n
(zq)n =
= r+1Fr
[
a1, q
√
a1,−q
√
a1, q
√
a1/p,−q
√
a1p, a6, . . . , ar+1; q, p;−z
√
a1,−
√
a1,
√
a1p,−
√
a1/p, a1q/a6, . . . , a1q/ar+1
]
. (1.5)
A truncated very well-poised theta hypergeometric series is defined by
r+1Vr[a1; a6, . . . , ar+1; q, p; z]N =
N∑
n=0
θ(a1q
2n; p)[a1, a6, . . . , ar+1; q, p; ]n
θ(a1; p)[a1q/a6, . . . , a1q/ar+1; q, p]n
(zq)n. (1.6)
We call a series of the form
n∑
k=−m
[a1, a2, . . . , ar+1; q, p]kz
k
[q, b1, b2, . . . , br; q, p]k
a finite bilateral theta hypergeometric series.
We shall make use of the following identity:
n∑
k=−m
λk+m
n−k∑
j=0
Aj =
n∑
k=−m
Ak+m
n−k∑
j=0
λj . (1.7)
Proof of (1.7). In order to prove (1.7) let us consider the following well know identity:
n∑
k=0
λk
n−k∑
j=0
Aj =
n∑
k=0
Ak
n−k∑
j=0
λj (1.8)
(cf. Gasper, Rahman [1, p. 321], (11.6.18)). Taking n+m for n and replacing k by k +m in (1.8),
we get (1.7) after some simplification.
Following summations are also needed in our analysis,
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
, (1.9)
where bcdeq−n = a2q (cf. Gasper, Rahman [1, p. 321], (11.4.1)). Now, setting e = aqn+1 in (1.9),
we get
8V7[a; b, c, a/bc; q, p]n =
[aq, aq/bc, bq, cq; q, p]n
[q, aq/b, aq/c, bcq; q, p]n
(1.10)
(cf. Gasper, Rahman [1, p. 322], (11.4.10)). Again, we have
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 7
996 R. Y. DENIS, S. N. SINGH, S. P. SINGH
10V9[dp; a, b, dpq/c, cdpq
n/ab, q−n; q, p2] =
[dpq, c/a, c/b, dpq/ab; q, p2]n
[c/ab, dpq/a, c, dpq/b; q, p2]n
(1.11)
(cf. Gasper, Rahman [1, p. 323], (11.4.11)). Now, taking c = abq in (1.11), we get
8V7[dp; a, b, dp/ab; q, p
2]n =
[dpq, aq, bq, dpq/ab; q, p2]n
[q, abq, dpq/a, dpq/b; q, p2]n
. (1.12)
We also have
n∑
k=0
θ{ad(rst/q)k, (b/d)(r/q)k, (c/d)(s/q)k, (ad/bc)(t/q)k; p}
θ(ad, b/d, c/d, ad/bc; p)
×
× [a; rst/q2, p]k[b; r, p]k[c; s, p]k[ad
2/bc; t, p]kq
k
[dq; q, p]k[adst/bq; st/q, p]k[adrt/cq; rt/q, p]k[bcrs/dq; rs/q, p]k
=
=
θ(a, b, c, ad2/bc; p)[arst/q2; srt/q2, p]n
dθ(ad, b/d, c/d, ad/bc; p)[dq; q, p]n[adst/bq; st/q, p]n
×
× [br; r, p]n[cs; s, p]n[ad
2t/bc; t, p]n
[adrt/cq; rt/q, p]n[bcrs/dq; rs/q, p]n
−
− θ(d, ad/b, ad/c, bc/d; p)
dθ(ad, b/d, /d, ad/bc; p)
(1.13)
(cf. Gasper, Rahman [1, p. 327], (11.6.9)). Taking d = 1 in the above, we get
n∑
k=0
θ{a(rst/q)k, (b)(r/q)k, (c)(s/q)k, (a/bc)(t/q)k; p}
θ(a, b, c, a/bc; p)
×
× [a; rst/q2, p]k[b; r, p]k[c; s, p]k[a/bc; t, p]kq
k
[q; q, p]k[ast/bq; st/q, p]k[art/cq; rt/q, p]k[bcrs/q; rs/q, p]k
=
=
[arst/q2; srt/q2, p]n[br; r, p]n[cs; s, p]n[at/bc; t, p]n
[q; q, p]n[ast/bq; st/q, p]n[art/cq; rt/q, p]n[bcrs/dq; rs/q, p]n
. (1.14)
2. Main results. In this section we shall establish our main transformations. We start by setting
λk =
θ(aq2k; p)[a, b, c, a/bc; q, p]kq
k
θ(a; p)[q, aq/b, aq/c, bcq; q, p]k
and
Ak =
θ(αp1q
2k
1 ; p21)[αp1, β, γ, αp1/βγ; q1, p
2
1]kq
k
1
θ(αp1; p21)[q1, αp1q1/β, αp1q1/γ, βγq1; q1, p
2
1]k
in (1.7) and using (1.10) and (1.12), we get
θ(aq2m; p)[a, b, c, a/bc; q, p]m[αp1q1, αp1q1/βγ, βq1, γq1; q1, p
2
1]nq
m
θ(α; p)[q, aq/b, aq/c, bcq; q, p]m[αp1q1/β, αp1q1/γ, βγq1; q1, p21]n
×
×
n∑
k=−m
θ(aq2m+2k; p)[aqm, bqm, cqm, aqm/bc; q, p]kq
k
θ(aq2m; p)[q1+m, aq1+m/b, aq1+m/c, bcq1+m; q, p]k
×
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 7
ON TRANSFORMATION FORMULAE FOR THETA HYPERGEOMETRIC FUNCTIONS 997
× [q−n1 , βq−n1 /αp1, γq
−n
1 /αp1, q
−n
1 /βγ; q1, p
2
1]k
[q−n1 /αp1, q
−n
1 /β, q−n1 /γ, βγq−n1 /αp1; q1; p21]k
=
=
θ(αp1q
2m
1 ; p21)[αp1, β, γ, αp1/βγ; q1, p
2
1]m[aq, bq, cq, aq/bc; q, p]nq
m
1
θ(αp1; p21)[q1, αp1q1/β, αp1q1/γ, βγq1; q1, p
2
1]m[q, aq/b, aq/c, bcq; q, p]n
×
×
n∑
k=−m
θ(αp1q
2m+2k
1 ; p21)[αp1q
m
1 , βq
m
1 , γq
m
1 , αp1q
m
1 /βγ; q1, p
2
1]kq
k
1
θ(αp1q2m1 ; p21)[q
1+m
1 , αp1q
1+m
1 /β, αp1q
1+m
1 /γ, βγq1+m
1 ; q1, p21]k
×
× [q−n, bq−n/a, cq−n/a, q−n/bc; q, p]k
[q−n/a, q−n/b, q−n/c, bcq−n/a; q, p]k
. (2.1)
Next, putting
λk =
θ(aq2k; p)[a, b, c, a/bc; q, p]kq
k
θ(a; p)[q, aq/b, aq/c, bcq; q, p]k
and
Ak =
θ{α(rst/q1)k, β(r/q1)k, γ(s/q1)k, (α/βγ)(t/q1)k; p1}
θ(α, β, γ, α/βγ; p1)
×
× [α; rst/q21, p1]k[β; r, p1]k[γ; s, p1]k[α/βγ; t, p1]kq
k
1
[q1; q1, p1]k[αst/βq1; st/q1, p1]k[αrt/γq1; rt/q1, p1]k[βγrs/q1; rs/q1, p1]k
in (1.7) and using (1.10) and (1.14), we get
θ(aq2m; p)[a, b, c, a/bc; q, p]m[αrst/q21; rst/q
2
1, p1]n[βr; r, p1]n
θ(a; p)[q, aq/b, aq/c, bcq; q, p]m[q1; q1, p1]n[αst/βq1; st/q1, p1]n
×
× [γs; s, p1]n[αt/βγ; t, p1]nq
m
[αrt/γq1; rt/q1, p1]n[βγrs/q1; rs/q1, p1]n
×
×
n∑
k=−m
θ(aq2m+2k; p)[aqm, bqm, cqm, aqm/bc; q, p]kq
k
θ(aq2m; p)[q1+m, aq1+m/b, aq1+m/c, bcq1+m; q, p]k
×
× [q−n1 ; q1, p1]k[β(st/q1)
−n/α; st/q1, p1]k
[(rst/q21)
−n/α; rst/q21, p1]k[r
−n/β; r, p1]k
×
× [γ(rt/q1)
−n/α; rt/q1, p1]k[(rs/q1)
−n/βγ; rs/q1, p1]k
[s−n/γ; s, p1]k[βγt−n/α; t, p1]k
=
=
[α; rst/q21, p1]m[β; r, p1]m[γ; s, p1]m[α/βγ; t, p1]mq
m
1
[αst/βq1; st/q1, p1]n[q1; q1, p1]m[αrt/γq1; rt/q1, p1]m[βγrs/q1; rs/q1, p1]m
×
× [aq, aq/bc, bq, cq; q, p]n
[q, aq/b, aq/c, bcq; q, p]n
×
×
n∑
k=−m
θ{α(rst/q1)k+m, β(r/q1)
k+m, γ(s/q1)
k+m, α(t/q1)
k+m/βγ; p1}
θ(α, β, γ, α/βγ; p1)
×
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 7
998 R. Y. DENIS, S. N. SINGH, S. P. SINGH
× [α(rst/q21)
m; rst/q21, p1]k[βr
m; r, p1]k[γs
m; s, p1]k
[q1+m
1 ; q1, p1]k[α(st/q1)1+m/β; st/q1, p1]k[α(rt/q1)m+1/γ; rt/q1, p1]k
×
× [αtm/βγ; t, p1]k[q
−n, bq−n/a, cq−n/a, q−n/bc; q, p]k
[βγ(rs/q1)m+1; rs/q1, p1]k[q−n/a, q−n/b, q−n/c, bcq−n/a; q, p]k
. (2.2)
Next, if we put
λk =
θ(apq2k; p2)[ap, b, c, ap/bc; q, p2]kq
k
θ(ap; p2)[q, apq/b, apq/c, bcq; q, p2]k
and
Ak =
θ{α(rst/q1)k, β(r/q1)k, γ(s/q1)k, α(t/q1)k/βγ; p1}
θ(α, β, γ, α/βγ; p1)
×
× [α; rst/q21, p1]k[β; r, p1]k[γ; s, p1]k[α/βγ; t, p1]kq
k
1
[q1; q1, p1]k[αst/βq1; st/q1, p1]k[αrt/γq1; rt/q1, p1]k[βγrs/q1; rs/q1, p1]k
in (1.7) and using (1.12) and (1.14), we get
θ(apq2m; p2)[ap, b, c, ap/bc; q, p2]mq
m
θ(ap; p2)[q, apq/b, apq/c, bcq; q, p2]m
×
× [αrst/q21; rst/q
2
1, p1]n[βr; r, p1]n[γs; s, p1]n[αt/βγ; t, p1]n
[q1; q1, p1]n[αst/βq1; st/q1, p1]n[αrt/γq1; rt/q1, p1]n[βγrs/q1; rs/q1, p1]n
×
×
n∑
k=−m
θ(apq2m+2k; p2)[apqm, bqm, cqm, apqm/bc; q, p]kq
k
θ(apq2m; p2)[q1+m, apq1+m/b, apq1+m/c, bcq1+m; q, p]k
×
× [q−n1 ; q1, p1]k[β(st/q1)
−n/α; st/q1, p1]k[γ(rt/q1)
−n/α; rt/q1, p1]k
[(rst/q21)
−n/α; rst/q21, p1]k[r
−n/β; r, p1]k[s−n/γ; s, p1]k
×
× [(rs/q1)
−n/βγ; rs/q1, p1]k
[βγt−n/α; t, p1]k
=
=
[α; rst/q21, p1]m[β; : r, p1]m[γ; s, p1]m[α/βγ; t, p1]mq
m
1
[q1; q1, p1]m[αst/βq1; st/q1, p1]m[αrt/γq1; rt/1, p1]m[βγrs/q1; rs/q1, p1]m
×
× [aq, aq/bc, bq, cq; q, p]n
[q, aq/b, aq/c, bcq; q, p]n
×
×
n∑
k=−m
θ{α(rst/q1)k+m, β(r/q1)
k+m, γ(s/q1)
k+m, α(t/q1)
k+m/βγ; p1}
θ(α, β, γ, α/βγ; p1)
×
× [α(rst/q21)
m; rst/q21; p1]k[βr
m; r, p1]k[γs
m; s, p1]k
[q1+m
1 , q1, p1]k[α(st/q1)1+m/β; st/q1, p1]k[α(rt/q1)1+m/γ; rt/q1, p1]k
×
× [αtm/βγ; t, p1]k[q
−n, bq−n/ap, cq−n/ap, q−n/bc; q, p2]k
[βγ(rs/q1)1+m; rs/q1, p1]k[q−n/ap, q−n/b, q−n/c, bcq−n/ap; q, p]k
. (2.3)
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 7
ON TRANSFORMATION FORMULAE FOR THETA HYPERGEOMETRIC FUNCTIONS 999
3. Special cases. In this section we shall deduce certain interesting special cases of our results.
If we set r = s = t = q1 in (2.2), we get after some simplification
θ(aq2m; p)[a, b, c, a/bc; q, p]m[αq1αq1/βγ, βq1, γq1; q1; p1]nq
m
θ(a; p)[q, aq/b, aq/c, bcq; q, p]m[αq1/β, αq1/γ, βγq1; q1, p1]n
×
×
n∑
k=−m
θ(aq2m+2k; p)[aqm, bqm, cqm, aqm/bc; q, p]kq
k
θ(aq2m; p)[q1+m, aq1+m/b, aq1+m/c, bcq1+m; q, p]k
×
× [q−n1 , βq−n1 /α, γq−n1 /α, q−n1 /βγ; q1, p1]k
[q−n1 /α, q−n1 /β, q−n1 /γ, βγq−n1 /α; q1, p1]k
=
=
[α, β, γ, α/βγ; q1, p1]mq
m
1 [aq, bq, cq, aq/bc; q, p]n
[q1, αq1/β, αq1/γ, βγq1; q1, p1]m[q, aq/b, aq/c, bcq; q, p]n
×
×
n∑
k=−m
θ(αq2m+2k
1 ; p1)[αq
m
1 , βq
m
1 , γq
m
1 , αq
m
1 /βγ; q1, p1]kq
k
1
θ(α; p1)[q
1+m
1 , αq1+m
1 /β, αq1+m
1 /γ, βγq1+m
1 ; q1, p1]k
×
× [q−n, bq−n/a, cq−n/a, q−n/bc; q, p]k
[q−n/a, q−n/b, q−n/c, bcq−n/a; q, p]k
. (3.1)
Now, setting α = βγ in (3.1), we get
n∑
k=0
θ(aq2k; p)[a, b, c, a/bc; q, p]k
θ(a; p)[q, aq/b, aq/c, bcq; q, p]k
=
[aq, aq/bc, bq, cq; q, p]n
[q, aq/b, aq/c, bcq; q, p]n
, (3.2)
which is (1.9).
Again, if we take α = βγq1 in (3.1), we get
θ(aq2m; p)θ(βq1, γq1, βγq
1+n
1 , q1+n
1 ; p1)[a, b, c, a/bc; q, p]mq
m
θ(a; p)θ(q1, βγq1, βq
1+n
1 , γq1+n
1 ; p1)[q, aq/b, aq/c, bcq; q, p]m
×
×
n∑
k=−m
θ(aq2m+2k; p)[aqm, bqm, cqm, aqm/bc; q, p]kq
k
θ(aq2m; p)[q1+m, aq1+m/b, aq1+m/c, bcq1+m; q, p]k
×
× [q−n1 , q−n−11 /β, q−n−11 /γ, q−n1 /βγ; q1, p1]k
[q−n−11 /βγ, q−n1 /β, q−n1 /γ, q−n−1; q1, p1]k
=
=
[β, γ; q1, p1]mq
m
1 [aq, bq, cq, aq/bc; q, p]n
[γq21, βq
2
1; q1, p1]m[q, aq/b, aq/c, bcq; q, p]n
×
×
n∑
k=−m
θ(βγq2m+2k+1
1 ; p1)[βq
m
1 , γq
m
1 ; q1, p1]kq
k
1
θ(βγ; p1)[βq
2+m
1 , γq2+m
1 ; q1, p1]k
×
× [q−n, bq−n/a, cq−n/a, q−n/bc; q, p]k
[q−n/a, q−n/b, q−n/c, bcq−n/a; q, p]k
. (3.3)
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 7
1000 R. Y. DENIS, S. N. SINGH, S. P. SINGH
Next, if we put m = 0 in (3.1), we get
[αq1, βq1, γq1, αq1/βγ; q1, p1]n
[q1, αq1/β, αq1/γ, βγq1; q1, p1]n
×
×
n∑
k=0
θ(aq2k; p)[a, b, c, a/bc; q, p]k[q
−n
1 , βq−n1 /α, γq−n1 /α, q−n1 /βγ; q1, p1]kq
k
θ(a; p)[q, aq/b, aq/c, bcq; q, p]k[q
−n
1 /α, q−n1 /β, q−n1 /γ, βγq−n1 /α; q1, p1]k
=
=
[aq, bq, cq, aq/bc; q, p]n
[q, aq/b, aq/c, bcq; q, p]n
×
×
n∑
k=0
θ(aq2k1 ; p1)[α, β, γ, α/βγ; q1, p1]k[q
−n, bq−n/a, cq−n/a, q−n/bc; q, p]kq
k
1
θ(α; p1)[q1, αq1/β, αq1/γ, βγq1; q1, p1]k[q−n/a, q−n/b, q−n/c, bcq−n/a; q, p]k
. (3.4)
If we set p = p1 = 0 and q1 = q in (3.4), we get the following interesting transformation:
10φ9
[
a, q
√
a− q
√
a, b, c, a/bc, βq−n/α, γq−n/α, q−n/βγ, q−n; q; q
√
a,−
√
a, aq/b, aq/c, bcq, q−n/α, q−n/β, q−n/γ, βγq−n/α
]
=
=
[aq, aq/bc, bq, cq, αq/β, αq/γ, βγq; q]n
[aq/b, aq/c, bcq, αq, βq, γq, αq/βγ; q]n
×
×10φ9
[
α, q
√
α,−q
√
α, β, γ, α/βγ, bq−n/a, cq−n/a, q−n/bc, q−n; q; q
√
α,−
√
α, αq/β, αq/γ, βγq, q−n/a, q−n/b, q−n/c, bcq−n/a
]
. (3.5)
Now, letting β → 1 in (3.5) we get the following summation of a truncated very well poised 6φ5
6φ5
[
a, q
√
a,−q
√
a, b, c, a/bc; q; q
√
a,−
√
a, aq/b, aq/c, bcq
]
n
=
[aq/aq/bc, bq, cq; q]n
[q, aq/b, aq/c, bcq; q]n
.
It is evident that several other interesting results involving theta hypergeometric functions can be
established.
Acknowledgement. The authors express deep appreciation to the referee for his valuable sug-
gestions.This has certainly improved the quality of the paper. They are thankful to the Department
of Science and Technology, Govt. of India, New Delhi, for support under major research projects
No. SR/S4/MS-461/07 dtd.13.2.2008 entitled “A study of basic hypergeometric functions with spe-
cial reference to Ramanujan mathematics”; No. SR/S4/MS:524 dtd.10.2.2008 entitled “Glimpses of
Ramanujan’s mathematics in the field of q-series” and No. F.6-2(23)/2008(MRC/NRCB) dtd.5.6.2009,
entitled “Investigations of Ramanujan’s work in the field of basic hypergeometric series” sanctioned
to them, respectively. The first author is also thankful to the Indian Society of Mathematics and
Mathematical Sciences (ISMAMS) for sponsoring his project.
1. Gasper G., Rahman M. Basic hypergeometric series. – Second ed. – Cambridge Univ. Press, 2004.
2. Spiridonov V. P. Theta hypergeometric series // Proc. NATO ASI Asympt. Combin. Appl. Math. Phys. (St.Petersburg,
July 9 – 22, 2001). – Dordrecht: Kluwer Acad. Publ., 2002. – P. 307 – 327.
3. Spiridonov V. P. An elliptic incarnation of the Bailey chain // In. Math. Res. Not. – 2002. – 37. – P. 1945 – 1977.
Received 14.08.11,
after revision — 15.06.12
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 7
|
| id | umjimathkievua-article-2634 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:27:17Z |
| publishDate | 2012 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/43/8c4713e89d7fc9514c4a93af32214943.pdf |
| spelling | umjimathkievua-article-26342020-03-18T19:31:34Z On transformation formulae for theta hypergeometric functions Про формули перетворення для гiпергеометричних тета-функцiй Denis, R. Y. Singh, S. N. Singh, S. P. Деніс, Р. Ю. Сінгх, С. Н. Сінгх, С. П. Using an identity and certain summation formulas for truncated theta hypergeometric series, we establish transformation formulas for finite bilateral theta hypergeometric series. За допомогою однiєї тотожностi та формул пiдсумовування скорочених гiпергеометричних тета-рядiв встановлено формули перетворення для скiнченних двостороннiх гiпергеометричних тета-рядiв. Institute of Mathematics, NAS of Ukraine 2012-07-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2634 Ukrains’kyi Matematychnyi Zhurnal; Vol. 64 No. 7 (2012); 994-1000 Український математичний журнал; Том 64 № 7 (2012); 994-1000 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2634/2027 https://umj.imath.kiev.ua/index.php/umj/article/view/2634/2028 Copyright (c) 2012 Denis R. Y.; Singh S. N.; Singh S. P. |
| spellingShingle | Denis, R. Y. Singh, S. N. Singh, S. P. Деніс, Р. Ю. Сінгх, С. Н. Сінгх, С. П. On transformation formulae for theta hypergeometric functions |
| title | On transformation formulae for theta hypergeometric functions |
| title_alt | Про формули перетворення для гiпергеометричних тета-функцiй |
| title_full | On transformation formulae for theta hypergeometric functions |
| title_fullStr | On transformation formulae for theta hypergeometric functions |
| title_full_unstemmed | On transformation formulae for theta hypergeometric functions |
| title_short | On transformation formulae for theta hypergeometric functions |
| title_sort | on transformation formulae for theta hypergeometric functions |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2634 |
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