Well-posed reduction formulas for the q-Kampé-de-Fériet function
By using the limiting case of Watson’s q-Whipple transformation as n → ∞, we investigate the transformations of the nonterminating q-Kampé-de-Fériet series. Further, new formulas for the transformations and well-posed reduction formulas are established for the basic Clausen hypergeometric series. Se...
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irk-123456789-1662632020-02-20T01:26:11Z Well-posed reduction formulas for the q-Kampé-de-Fériet function Chu, W. Zhang, W. Статті By using the limiting case of Watson’s q-Whipple transformation as n → ∞, we investigate the transformations of the nonterminating q-Kampé-de-Fériet series. Further, new formulas for the transformations and well-posed reduction formulas are established for the basic Clausen hypergeometric series. Several remarkable formulas are also found for new function classes beyond the q-Kampé-de-Fériet function. За допомогою граничного випадку n→∞ для ватсонівського q-перетворення Віппла досліджено перетворення нескінченного q-ряду Кампе де Фер'є. Крім того, встановлено нові формули перетворень та коректні формули редукції для базового гіпергеометричного ряду Кпаузена. Декілька важливих формул знайдено також для нових класів функцій, до яких не належить q-функція Кампе де Фер'є. 2010 Article Well-posed reduction formulas for the q-Kampé-de-Fériet function / W. Chu, W. Zhang // Український математичний журнал. — 2010. — Т. 62, № 11. — С. 1538–1554. — Бібліогр.: 10 назв. — англ. 1027-3190 http://dspace.nbuv.gov.ua/handle/123456789/166263 517.9 en Український математичний журнал Інститут математики НАН України |
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| description |
By using the limiting case of Watson’s q-Whipple transformation as n → ∞, we investigate the transformations of the nonterminating q-Kampé-de-Fériet series. Further, new formulas for the transformations and well-posed reduction formulas are established for the basic Clausen hypergeometric series. Several remarkable formulas are also found for new function classes beyond the q-Kampé-de-Fériet function. |
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| author |
Chu, W. Zhang, W. |
| author_facet |
Chu, W. Zhang, W. |
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Chu, W. |
| title |
Well-posed reduction formulas for the q-Kampé-de-Fériet function |
| title_short |
Well-posed reduction formulas for the q-Kampé-de-Fériet function |
| title_full |
Well-posed reduction formulas for the q-Kampé-de-Fériet function |
| title_fullStr |
Well-posed reduction formulas for the q-Kampé-de-Fériet function |
| title_full_unstemmed |
Well-posed reduction formulas for the q-Kampé-de-Fériet function |
| title_sort |
well-posed reduction formulas for the q-kampé-de-fériet function |
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Інститут математики НАН України |
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2010 |
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Статті |
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| citation_txt |
Well-posed reduction formulas for the q-Kampé-de-Fériet function / W. Chu, W. Zhang // Український математичний журнал. — 2010. — Т. 62, № 11. — С. 1538–1554. — Бібліогр.: 10 назв. — англ. |
| series |
Український математичний журнал |
| work_keys_str_mv |
AT chuw wellposedreductionformulasfortheqkampedeferietfunction AT zhangw wellposedreductionformulasfortheqkampedeferietfunction |
| first_indexed |
2025-07-14T21:05:04Z |
| last_indexed |
2025-07-14T21:05:04Z |
| _version_ |
1837657865281601536 |
| fulltext |
UDC 517.9
W. Chu, W. Zhang (School Math. Sci. Dalian Univ. Technology, China)
WELL-POSED REDUCTION FORMULAE
FOR q-KAMPÉ DE FÉRIET FUNCTION
КОРЕКТНI ФОРМУЛИ РЕДУКЦIЇ
ДЛЯ q-ФУНКЦIЇ КАМПЕ ДЕ ФЕР’Є
By means of the limiting case n→∞ of Watson’s q-Whipple transformation, we investigate transformations
on the nonterminating q-Kampé de Fériet series. Further new transformation and well-posed reduction formulae
are established for the basic Clausen hypergeometric series. Several remarkable formulae are found also for
new function classes beyond q-Kampé de Fériet function.
За допомогою граничного випадку n → ∞ для ватсонiвського q-перетворення Вiппла дослiджено пе-
ретворення нескiнченного q-ряду Кампе де Фер’є. Крiм того, встановлено новi формули перетворень та
коректнi формули редукцiї для базового гiпергеометричного ряду Клаузена. Декiлька важливих формул
знайдено також для нових класiв функцiй, до яких не належить q-функцiя Кампе де Фер’є.
1. Introduction and motivation. For the two indeterminates x and q, the shifted
factorial of x with base q reads as
(x; q)0 = 1 and (x; q)n = (1− x) (1− qx) . . . (1− qn−1x) for n ∈ N.
When |q| < 1, there are two well-defined infinite product expressions
(x; q)∞ =
∞∏
k=0
(1− qkx) and (x; q)n = (x; q)∞ / (qnx; q)∞ .
The product and fraction of shifted factorials are abbreviated respectively to
[ α, β, . . . , γ; q ]n = (α; q)n (β; q)n . . . (γ; q)n ,[
α, β, . . . , γ
A, B, . . . , C
∣∣∣∣∣ q
]
n
=
(α; q)n (β; q)n . . . (γ; q)n
(A; q)n (B; q)n . . . (C; q)n
.
Following Bailey [1] and Gasper, Rahman [5], the basic hypergeometric series is defined
by
1+rφs
[
a0, a1, . . . , ar
b1, . . . , bs
∣∣∣∣∣ q; z
]
=
=
∞∑
n=0
{
(−1)nq(
n
2)
}s−r [a0, a1, . . . , ar
q, b1, . . . , bs
∣∣∣∣∣ q
]
n
zn,
where the base q will be restricted to |q| < 1 for nonterminating q-series.
Among the q-series transformations, Watson’s one is fundamental (cf. [5], III-18)
8φ7
[
a, q
√
a, −q
√
a, b, c, d, e, q−n
√
a, −
√
a, qa/b, qa/c, qa/d, qa/e, qn+1a
∣∣∣∣∣ q; qn+2a2
bcde
]
= (1a)
c© W. CHU, W. ZHANG, 2010
1538 ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 11
WELL-POSED REDUCTION FORMULAE FOR q-KAMPÉ DE FÉRIET FUNCTION 1539
=
[
qa, qa/bc
qa/b, qa/c
∣∣∣∣∣ q
]
n
4φ3
[
q−n, b, c, qa/de
qa/d, qa/e, q−nbc/a
∣∣∣∣∣ q; q
]
. (1b)
Its limiting case n→∞ reads equivalently as the nonterminating transformation
3φ2
[
a, c, e
b, d
∣∣∣∣∣ q; bdace
]
=
[
bd/ae, bd/ce
bd/e, bd/ace
∣∣∣∣∣ q
]
∞
× (2a)
×
∞∑
i=0
(−1)i
1− q2i−1bd/e
1− bd/qe
q(
i
2)
[
bd/qe, a, c, b/e, d/e
q, bd/ae, bd/ce, d, b
∣∣∣∣∣ q
]
i
(
bd
ac
)i
.
(2b)
As the q-analogue of Kampé de Fériet function, Srivastava and Karlsson [10, p. 349]
define the generalized bivariate basic hypergeometric function by
Φλ:r;sµ:u;v
[
α1, . . . , αλ : a1, . . . , ar; c1, . . . , cs; q : x, y
β1, . . . , βµ : b1, . . . , bu; d1, . . . , dv; i, j, k
]
=
=
∞∑
m,n=0
[α1, . . . , αλ; q]m+n
[β1, . . . , βµ; q]m+n
[a1, . . . , ar; q]m[c1, . . . , cs; q]n
[b1, . . . , bu; q]m[d1, . . . , dv; q]n
xmynqi(
m
2 )+j(n
2)+kmn
(q; q)m(q; q)n
.
When i, j, k ∈ N0, this double series Φλ:r;sµ:u;v is convergent for |x| < 1, |y| < 1 and
|q| < 1. There has not been much attention to this series in the literature. By means
of the q-analogue of Kummer – Thomae – Whipple and the Hall transformation on 3φ2-
series (cf. [5], III-9 and III-10), Chu, Jia [3] and Jia, Wang [7] investigated systematically
summation and reduction formulae respectively for the Φ0:3;λ
1:1;µ and Φ1:2;λ
1:1;µ series. Chu,
Jia [2] and Chu, Srivastava [4] derived several transformation and reduction formulae
respectively by inversion techniques and formal power series method. Jeugt [6] deter-
mined invariant transformation group for the double Clausen series with λ+ r = 3 and
µ+ u = 2. Further works can be found in Jia, Wang [8] and Singh [9] as well as those
cited by Chu, Jia [3].
The purpose of this paper is to investigate the above defined nonterminating q-
Kampé de Fériet function exclusively by employing the limiting transformation (2a),
(2b). The rest of the paper will be divided into five sections, devoted respectively to the
five series labeled by
Φ0:3;λ
1:1;µ, Φ2:1;λ
2:0;µ, Φ1:2;λ
2:0;µ, Φ1:2;λ
1:1;µ, Φ0:3;λ
2:0;µ.
Several transformation and well-posed reduction formulae on these five series will be
established which can briefly be commented as follows:
Most of the reduction formulae displayed from Section 2 to Section 4 are well-posed,
unlike the usual ones appeared in Chu, Jia [3] and Jia, Wang [7].
Even though the two double series Φ0:3;λ
1:1;µ and Φ1:2;λ
1:1;µ have intensively been studied
by Chu, Jia [3] and Jia, Wang [7], the results shown in Section 2 and Section 5 are
substantially different.
It is remarkable that there exist reduction formulae for new function classes beyond
the q-Kampé de Fériet function, which are exemplified for the two transformations
proven in Sections 5 and 6.
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 11
1540 W. CHU, W. ZHANG
2. Nonterminating double series Φ0:3;λ
1:1;µ. By means of the two transformations
for 3φ2-series (cf. [5], III-9 and III-10), the double Φ0:3;λ
1:1;µ series has intensively been
investigated by Chu, Jia [3] (§ 2), where numerous reduction and summation formulae
have been obtained. Instead, we shall utilize (2a), (2b) to transform this Φ0:3;λ
1:1;µ series
into Φ2:3;λ+1
3:1;µ+1 series modified by a “well-posed” factor, which yields four well-posed
reduction formulae for the former and two unusual ones for the latter.
Theorem 1. For an arbitrary sequence {Ω(j)}, there holds the transformation[
bd/e, bd/ace
bd/ae, bd/ce
∣∣∣∣∣ q
]
∞
∞∑
i,j=0
Ω(j)
(b; q)i+j
[
a, c, e
q, d
∣∣∣∣∣ q
]
i
(
qjbd
ace
)i
= (3a)
=
∞∑
i,j=0
1− q2i+j−1bd/e
1− bd/qe
[
b/e, bd/qe
b, bd/ae, bd/ce
∣∣∣∣∣ q
]
i+j
× (3b)
×q(
i
2)+ij
[
a, c, d/e
q, d
∣∣∣∣∣ q
]
i
(
−bd
ac
)i [bd/ace
b/e
∣∣∣∣∣ q
]
j
Ω(j) (3c)
provided that both double series displayed above are absolutely convergent.
Proof. The theorem follows directly by writing the double sum in (3a) as
∞∑
i,j=0
Ω(j)
(b; q)i+j
[
a, c, e
q, d
∣∣∣∣∣ q
]
i
(
qjbd
ace
)i
=
∞∑
j=0
Ω(j)
(b; q)j
3φ2
[
a, c, e
qjb, d
∣∣∣∣∣ q; qjbdace
]
(4)
and then transforming, via (2a), (2b), the 3φ2-series into
3φ2
[
a, c, e
qjb, d
∣∣∣∣∣ q; qjbdace
]
=
[
qjbd/ae, qjbd/ce
qjbd/e, qjbd/ace
∣∣∣∣∣ q
]
∞
∞∑
i=0
(−1)i
1− q2i+j−1bd/e
1− qj−1bd/e
×
×q(
i
2)+ij
[
qj−1bd/e, a, c, qjb/e, d/e
q, qjbd/ae, qjbd/ce, d, qjb
∣∣∣∣∣ q
]
i
(
bd
ac
)i
=
=
[
bd/ae, bd/ce
bd/e, bd/ace
∣∣∣∣∣ q
]
∞
[
b, bd/ace
b/e
∣∣∣∣∣ q
]
j
∞∑
i=0
(−1)i
[
a, c, d/e
q, d
∣∣∣∣∣ q
]
i
×
×1− q2i+j−1bd/e
1− bd/qe
q(
i
2)+ij
[
b/e, bd/qe
b, bd/ae, bd/ce
∣∣∣∣∣ q
]
i+j
(
bd
ac
)i
.
Theorem 1 is proved.
Instead, applying the q-analogue of Kummer – Thomae – Whipple transformation (cf. [5],
III-9), we can reformulate the 3φ2-series displayed in (4) as
3φ2
[
a, c, e
qjb, d
∣∣∣∣∣ q; qjbdace
]
=
[
d/a, qjbd/ce
d, qjbd/ace
∣∣∣∣∣ q
]
∞
3φ2
[
a, qjb/c, qjb/e
qjb, qjbd/ce
∣∣∣∣∣ q; da
]
where the 3φ2-series on the right-hand side of the last equation can further be restated
by means of (2a), (2b) as
3φ2
[
a, qjb/c, qjb/e
qjb, qjbd/ce
∣∣∣∣∣ q; da
]
=
[
qjbd/ac, qjbd/ae
d/a, q2jb2d/ace
∣∣∣∣∣ q
]
∞
∞∑
i=0
1− q2i+2j−1b2d/ace
1− q2j−1b2d/ace
×
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 11
WELL-POSED REDUCTION FORMULAE FOR q-KAMPÉ DE FÉRIET FUNCTION 1541
×(−1)iq(
i
2)
[
q2j−1b2d/ace, qjb/a, qjb/c, qjb/e, qjbd/ace
q, qjb, qjbd/ac, qjbd/ae, qjbd/ce
∣∣∣∣∣ q
]
i
di.
Substituting them successively into (4), we find the following alternative expression:[
d, bd/ace, b2d/ace
bd/ac, bd/ae, bd/ce
∣∣∣∣∣ q
]
∞
∞∑
i,j=0
Ω(j)
(b; q)i+j
[
a, c, e
q, d
∣∣∣∣∣ q
]
i
(
qjbd
ace
)i
=
=
∞∑
j=0
Ω(j)
[b/a, b/c, b/e; q]j
∞∑
i=0
(−1)i
1− q2i+2j−1b2d/ace
1− b2d/qace
q(
i
2)×
×
[
b/a, b/c, b/e, bd/ace
b, bd/ac, bd/ae, bd/ce
∣∣∣∣∣ q
]
i+j
(b2d/qace; q)i+2j
(q; q)i
di.
Letting n := i+ j and then keeping in mind of the relation
(b2d/qace; q)n+j
(q; q)n−j
= (−1)jqjn−(j
2) (b2d/qace; q)n
(q; q)n
[
q−n, qn−1b2d/ace; q
]
j
we can equivalently reformulate Theorem 1 as another transformation.
Theorem 2. For an arbitrary sequence {Ω(j)}, there holds the transformation[
d, bd/ace, b2d/ace
bd/ac, bd/ae, bd/ce
∣∣∣∣∣ q
]
∞
∞∑
i,j=0
Ω(j)
(b; q)i+j
[
a, c, e
q, d
∣∣∣∣∣ q
]
i
(
qjbd
ace
)i
= (5a)
=
∞∑
n=0
1− q2n−1b2d/ace
1− b2d/qace
[
b/a, b/c, b/e, bd/ace, b2d/qace
q, b, bd/ac, bd/ae, bd/ce
∣∣∣∣∣ q
]
n
× (5b)
×(−1)nq(
n
2)dn
n∑
j=0
( q
d
)j [q−n, qn−1b2d/ace
b/a, b/c, b/e
∣∣∣∣∣ q
]
j
Ω(j) (5c)
provided that both double series displayed above are absolutely convergent.
According to Theorems 1 and 2, we are now going to derive five reduction transfor-
mation formulae by concretely specifying Ω(j) in five different manners.
2.1. For the Ω(j) sequence given by
Ω(j) =
[
α, b/a, b/c, b/e
q, w, b2dα/wace
∣∣∣∣∣ q
]
j
dj
evaluating the inner sum with respect to j in (5) by means of the q-Pfaff – Saalschütz
theorem (cf. [5], II-12)
3φ2
[
q−n, a, b
c, q1−nab/c
∣∣∣∣∣ q; q
]
=
[
c/a, c/b
c, c/ab
∣∣∣∣∣ q
]
n
(6)
and then simplifying the corresponding equation displayed in Theorem 2, we obtain the
following reduction formula.
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 11
1542 W. CHU, W. ZHANG
Proposition 1 (well-posed reduction formula).
∞∑
i,j=0
qijdj
(b; q)i+j
[
a, c, e
q, d
∣∣∣∣∣ q
]
i
[
α, b/a, b/c, b/e
q, w, b2dα/wace
∣∣∣∣∣ q
]
j
(
bd
ace
)i
=
=
[
bd/ac, bd/ae, bd/ce
d, bd/ace, b2d/ace
∣∣∣∣∣ q
]
∞
∞∑
n=0
(−1)nq(
n
2) 1− q2n−1b2d/ace
1− b2d/qace
×
×
[
b/a, b/c, b/e, w/α, bd/ace, b2d/qace, b2d/wace
q, b, w, bd/ac, bd/ae, bd/ce, b2dα/wace
∣∣∣∣∣ q
]
n
(dα)n.
The limiting case α→∞ of this proposition yields an interesting transformation.
Corollary 1 (well-posed reduction formula).
∞∑
i,j=0
qij
(b; q)i+j
[
a, c, e
q, d
∣∣∣∣∣ q
]
i
[
b/a, b/c, b/e
q, w
∣∣∣∣∣ q
]
j
(
bd
ace
)i (wace
b2
)j
=
=
[
bd/ac, bd/ae, bd/ce
d, bd/ace, b2d/ace
∣∣∣∣∣ q
]
∞
×
×8φ7
b2d
qace
,
√
qb2d
ace
, −
√
qb2d
ace
,
b
a
,
b
c
,
b
e
,
bd
ace
,
b2d
wace√
b2d
qace
, −
√
b2d
qace
, b, w,
bd
ac
,
bd
ae
,
bd
ce
∣∣∣∣∣∣∣∣∣ q;
wace
b2
.
2.2. For the Ω(j) sequence defined by
Ω(j) = q(
j
2) [b/a, b/c, b/e; q]j
(q; q)j(b2d/ace; q2)j
dj
evaluating the inner sum with respect to j displayed in (5) through the q-analogue of
Gauss’ 2F1
(
1
2
)
sum (cf. [5], II-11)
2φ2
[
a, b
√
qab, −
√
qab
∣∣∣∣∣ q;−q
]
=
[
qa, qb
q, qab
∣∣∣∣∣ q2
]
∞
(7)
and then simplifying the corresponding equation in Theorem 2, we get the following
reduction formula.
Proposition 2 (well-posed reduction formula).
∞∑
i,j=0
q(
j
2)
[
a, c, e
q, d
∣∣∣∣∣ q
]
i
(qid)j [b/a, b/c, b/e; q]j
(b; q)i+j(q; q)j(b2d/ace; q2)j
(
bd
ace
)i
=
=
[
bd/ac, bd/ae, bd/ce
d, bd/ace, b2d/ace
∣∣∣∣∣ q
]
∞
∞∑
n=0
(−1)nq(
n
2) 1− q4n−1b2d/ace
1− b2d/qace
×
×
[
b/a, b/c, b/e, bd/ace
b, bd/ac, bd/ae, bd/ce
∣∣∣∣∣ q
]
2n
(b2d/qace; q2)n
(q2; q2)n
d2n.
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 11
WELL-POSED REDUCTION FORMULAE FOR q-KAMPÉ DE FÉRIET FUNCTION 1543
2.3. For the Ω(j) sequence specified by
Ω(j) =
[
b/a, b/c, b/e
q, w
∣∣∣∣∣ q
]
j
(w; q2)j
(b2d/ace; q2)j
dj
evaluating the inner sum with respect to j displayed in (5) via Andrews’ terminating
q-analogue of the Watson 3F2-sum (cf. [5], II-17)
4φ3
[
q−n, qn+1a2, c, −c
c2, qa, −qa
∣∣∣∣∣ q; q
]
=
=
0, n− odd,
cn
[
q, q2a2/c2
q2a2, qc2
∣∣∣∣∣ q2
]
m
, n = 2m,
(8)
and then simplifying the corresponding equation in Theorem 2, we find the following
reduction formula.
Proposition 3 (well-posed reduction formula).
∞∑
i,j=0
(qid)j
(b; q)i+j
[
a, c, e
q, d
∣∣∣∣∣ q
]
i
[
b/a, b/c, b/e
q, w
∣∣∣∣∣ q
]
j
(w; q2)j
(b2d/ace; q2)j
(
bd
ace
)i
=
=
[
bd/ac, bd/ae, bd/ce
d, bd/ace, b2d/ace
∣∣∣∣∣ q
]
∞
∞∑
n=0
q(
2n
2 ) 1− q4n−1b2d/ace
1− b2d/qace
×
×
[
b/a, b/c, b/e, bd/ace
b, bd/ac, bd/ae, bd/ce
∣∣∣∣∣ q
]
2n
[
b2d/qace, b2d/wace
q2, qw
∣∣∣∣∣ q2
]
n
(d2w)n.
2.4. In Theorem 1, rewrite the double sum (3a) as
∞∑
i=0
[
a, c, e
q, b, d
∣∣∣∣∣ q
]
i
(
bd
ace
)i ∞∑
j=0
Ω(j)
(qib; q)j
qij . (9)
Specializing the Ω(j) sequence explicitly by
Ω(j) =
[
β, γ
q
∣∣∣∣∣ q
]
j
(
b
βγ
)j
and then evaluating the sum with respect to j displayed in (9) by means of the q-Gauss
summation theorem (cf. [5], II-8)
2φ1
[
a, b
c
∣∣∣∣∣ q; cab
]
=
[
c/a, c/b
c, c/ab
∣∣∣∣∣ q
]
∞
(10)
we have from Theorem 1 the following interesting reduction formula.
Proposition 4 (reduction formula).
∞∑
i,j=0
1− q2i+j−1bd/e
1− bd/qe
[
b/e, bd/qe
b, bd/ae, bd/ce
∣∣∣∣∣ q
]
i+j
×
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 11
1544 W. CHU, W. ZHANG
×q(
i
2)
[
a, c, d/e
q, d
∣∣∣∣∣ q
]
i
(
−bd
ac
)i [β, γ, bd/ace
q, b/e
∣∣∣∣∣ q
]
j
(
qib
βγ
)j
=
=
[
b/β, b/γ, bd/e, bd/ace
b, b/βγ, bd/ae, bd/ce
∣∣∣∣∣ q
]
∞
4φ3
[
a, c, e, b/βγ
d, b/β, b/γ
∣∣∣∣∣ q; bdace
]
.
2.5. Alternatively, specializing the Ω(j) sequence explicitly by
Ω(j) = q(
j
2) [β, q/β; q]j
(q2; q2)j
bj
and then evaluating the sum with respect to j displayed in (9) through the q-analogue
of Bailey’s 2F1
(
1
2
)
sum (cf. [5], II-10)
2φ2
[
a, q/a
−q, b
∣∣∣∣∣ q;−b
]
=
[ab, qb/a; q2]∞
(b; q)∞
(11)
we derive from Theorem 1 another strange reduction formula.
Proposition 5 (reduction formula).
∞∑
i,j=0
1− q2i+j−1bd/e
1− bd/qe
[
b/e, bd/qe
b, bd/ae, bd/ce
∣∣∣∣∣ q
]
i+j
(
−bd
ac
)i
×
×q(
i+j
2 )
[
a, c, d/e
q, d
∣∣∣∣∣ q
]
i
[
β, q/β, bd/ace
q, −q, b/e
∣∣∣∣∣ q
]
j
bj =
=
[
b/β, bd/e, bd/ace
b, bd/ae, bd/ce
∣∣∣∣∣ q
]
∞
∞∑
i=0
[
a, c, e
q, d, b/β
∣∣∣∣∣ q
]
i
[
qibβ
qib/β
∣∣∣∣∣ q2
]
∞
(
bd
ace
)i
.
3. Nonterminating double series Φ2:1;λ
2:0;µ. There is a theorem connecting the Φ0:3;λ
1:1;µ
series in the last section to Φ2:1;λ
2:0;µ+1 series due to Chu, Jia [3] (Theorem 2.2), where no
reduction formulae was given for this last series. By means of (2a), (2b), this section will
prove a transformation theorem for this Φ2:1;λ
2:0;µ series and then derive four well-posed
reduction formulae.
Theorem 3. For an arbitrary sequence {Ω(j)}, there holds the transformation[
bd/e, bd/ace
bd/ae, bd/ce
∣∣∣∣∣ q
]
∞
∞∑
i,j=0
(
bd
ace
)i [a, c
b, d
∣∣∣∣∣ q
]
i+j
[
e
q
∣∣∣∣∣ q
]
i
Ω(j) = (12a)
=
∞∑
n=0
1− q2n−1bd/e
1− bd/qe
[
a, c, b/e, d/e, bd/qe
q, b, d, bd/ae, bd/ce
∣∣∣∣∣ q
]
n
× (12b)
×(−1)nq(
n
2)
(
bd
ac
)n n∑
j=0
(qac
bd
)j [q−n, qn−1bd/e
b/e, d/e
∣∣∣∣∣ q
]
j
Ω(j) (12c)
provided that both double series displayed above are absolutely convergent.
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 11
WELL-POSED REDUCTION FORMULAE FOR q-KAMPÉ DE FÉRIET FUNCTION 1545
Proof. Expressing the double sum in (12a) as
∞∑
i,j=0
(
bd
ace
)i [a, c
b, d
∣∣∣∣∣ q
]
i+j
[
e
q
∣∣∣∣∣ q
]
i
Ω(j) = (13a)
=
∞∑
j=0
Ω(j)
[
a, c
b, d
∣∣∣∣∣ q
]
j
3φ2
[
qja, qjc, e
qjb, qjd
∣∣∣∣∣ q; bdace
]
(13b)
and then transforming, by (2a), (2b), the above 3φ2-series into
3φ2
[
qja, qjc, e
qjb, qjd
∣∣∣∣∣ q; bdace
]
=
[
qjbd/ae, qjbd/ce
q2jbd/e, bd/ace
∣∣∣∣∣ q
]
∞
∞∑
i=0
1− q2i+2j−1bd/e
1− q2j−1bd/e
×
×(−1)iq(
i
2)
[
q2j−1bd/e, qja, qjc, qjd/e, qjb/e
q, qjbd/ae, qjbd/ce, qjb, qjd
∣∣∣∣∣ q
]
i
(
bd
ac
)i
=
=
[
bd/ae, bd/ce
bd/e, bd/ace
∣∣∣∣∣ q
]
∞
[
b, d
a, c, b/e, d/e
∣∣∣∣∣ q
]
j
∞∑
i=0
1− q2i+2j−1bd/e
1− bd/qe
×
×(−1)iq(
i
2)
[
a, c, b/e, d/e
b, d, bd/ae, bd/ce
∣∣∣∣∣ q
]
i+j
(bd/qe; q)i+2j
(q; q)i
(
bd
ac
)i
we derive the following equality:
Eq(12a) =
∞∑
j=0
Ω(j)
[b/e, d/e; q]j
∞∑
i=0
(−1)i
1− q2i+2j−1bd/e
1− bd/qe
q(
i
2)×
×
[
a, c, b/e, d/e
b, d, bd/ae, bd/ce
∣∣∣∣∣ q
]
i+j
(bd/qe; q)i+2j
(q; q)i
(
bd
ac
)i
.
Relabeling the summation indices by n := i+ j leads us to another expression
Eq(12a) =
∞∑
n=0
1− q2n−1bd/e
1− bd/qe
[
a, c, b/e, d/e
b, d, bd/ae, bd/ce
∣∣∣∣∣ q
]
n
×
×
n∑
j=0
Ω(j)
[b/e, d/e; q]j
(bd/qe; q)n+j
(q; q)n−j
(−1)n−jq(
n−j
2 )
(
bd
ac
)n−j
which is equivalent to the transformation in Theorem 3 in view of the relation
(bd/qe; q)n+j
(q; q)n−j
= (−1)jqjn−(j
2) (bd/qe; q)n
(q; q)n
[
q−n, qn−1bd/e; q
]
j
.
Theorem 3 is proved.
By specifying the Ω(j) sequence in terms of shifted factorial fractions, we shall
derive from Theorem 3 three reduction formulae.
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 11
1546 W. CHU, W. ZHANG
3.1. In Theorem 3, specializing the Ω(j) sequence firstly by
Ω(j) =
[
α, b/e, d/e
q, w, bdα/we
∣∣∣∣∣ q
]
j
(
bd
ac
)j
and then evaluating the inner sum with respect to j displayed in (12c) by means of (6)
lead us to the following reduction formula.
Proposition 6 (well-posed reduction formula).[
bd/e, bd/ace
bd/ae, bd/ce
∣∣∣∣∣ q
]
∞
∞∑
i,j=0
[
a, c
b, d
∣∣∣∣∣ q
]
i+j
[
e
q
∣∣∣∣∣ q
]
i
×
×
[
α, b/e, d/e
q, w, bdα/we
∣∣∣∣∣ q
]
j
(
bd
ace
)i(
bd
ac
)j
=
=
∞∑
n=0
(−1)nq(
n
2) 1−q2n−1bd/e
1− bd/qe
×
×
[
a, c, b/e, d/e, bd/qe, w/α, bd/we
q, b, d, w, bd/ae, bd/ce, bdα/we
∣∣∣∣∣ q
]
n
(
bdα
ac
)n
.
When α→∞, this proposition results in an interesting transformation.
Corollary 2 (well-posed reduction formula).[
bd/e, bd/ace
bd/ae, bd/ce
∣∣∣∣∣ q
]
∞
∞∑
i,j=0
[
a, c
b, d
∣∣∣∣∣ q
]
i+j
[
e
q
∣∣∣∣∣ q
]
i
×
×
[
b/e, d/e
q, w
∣∣∣∣∣ q
]
j
(
bd
ace
)i (we
ac
)j
=
= 8φ7
[
bd/qe, q
√
bd/qe, −q
√
bd/qe, a, c, b/e, d/e, bd/we√
bd/qe, −
√
bd/qe, b, d, w, bd/ae, bd/ce
∣∣∣∣∣ q; weac
]
.
3.2. In Theorem 3, specializing the Ω(j) sequence alternatively by
Ω(j) = q(
j
2) [b/e, d/e; q]j
(q; q)j(bd/e; q2)j
(
bd
ac
)j
and then evaluating the inner sum with respect to j displayed in (12c) by means of (7)
yield another reduction formula.
Proposition 7 (well-posed reduction formula).[
bd/e, bd/ace
bd/ae, bd/ce
∣∣∣∣∣ q
]
∞
∞∑
i,j=0
q(
j
2)
[
a, c
b, d
∣∣∣∣∣ q
]
i+j
[
e
q
∣∣∣∣∣ q
]
i
×
× [b/e, d/e; q]j
(q; q)j(bd/e; q2)j
(
bd
ace
)i(
bd
ac
)j
=
=
∞∑
n=0
(−1)nq(
n
2) 1−q4n−1bd/e
1− bd/qe
[
a, c, b/e, d/e
b, d, bd/ae, bd/ce
∣∣∣∣∣ q
]
2n
(bd/qe; q2)n
(q2; q2)n
(
bd
ac
)2n
.
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 11
WELL-POSED REDUCTION FORMULAE FOR q-KAMPÉ DE FÉRIET FUNCTION 1547
3.3. In Theorem 3, specializing the Ω(j) sequence finally by
Ω(j) =
[
b/e, d/e
q, w
∣∣∣∣∣ q
]
j
(w; q2)j
(bd/e; q2)j
(
bd
ac
)j
and then evaluating the inner sum with respect to j displayed in (12c) by means of (8)
give rise to the following reduction formula.
Proposition 8 (well-posed reduction formula).[
bd/e, bd/ace
bd/ae, bd/ce
∣∣∣∣∣ q
]
∞
∞∑
i,j=0
[
a, c
b, d
∣∣∣∣∣ q
]
i+j
[
e
q
∣∣∣∣∣ q
]
i
[
b/e, d/e
q, w
∣∣∣∣∣ q
]
j
×
× (w; q2)j
(bd/e; q2)j
(
bd
ace
)i(
bd
ac
)j
=
=
∞∑
n=0
q(
2n
2 ) 1−q4n−1bd/e
1− bd/qe
[
a, c, b/e, d/e
b, d, bd/ae, bd/ce
∣∣∣∣∣ q
]
2n
×
×
[
bd/qe, bd/we
q2, qw
∣∣∣∣∣ q2
]
n
(
bd
ac
)2n
wn.
4. Nonterminating double series Φ1:2;λ
2:0;µ. This section is devoted to the transfor-
mation and well-posed reduction formulae for the Φ1:2;λ
2:0;µ series, which does not seem to
have appeared previously in literature.
Theorem 4. For an arbitrary sequence {Ω(j)}, there holds the transformation[
bd/e, bd/ace
bd/ae, bd/ce
∣∣∣∣∣ q
]
∞
∞∑
i,j=0
(
qjbd
ace
)i [ a
b, d
∣∣∣∣∣ q
]
i+j
[
c, e
q
∣∣∣∣∣ q
]
i
Ω(j) = (14a)
=
∞∑
n=0
(−1)nq(
n
2) 1− q2n−1bd/e
1− bd/qe
[
a, c, b/e, d/e, bd/qe
q, b, d, bd/ae, bd/ce
∣∣∣∣∣ q
]
n
(
bd
ac
)n
×
(14b)
×
n∑
j=0
(−1)jq−(j
2)
(qa
bd
)j [ q−n, qn−1bd/e
q1−n/c, qnbd/ce
∣∣∣∣∣ q
]
j
[
bd/ace
b/e, d/e
∣∣∣∣∣ q
]
j
Ω(j) (14c)
provided that both double series displayed above are absolutely convergent.
Proof. Rewriting the double sum in (14a) as
∞∑
i,j=0
(
qjbd
ace
)i [ a
b, d
∣∣∣∣∣ q
]
i+j
[
c, e
q
∣∣∣∣∣ q
]
i
Ω(j) = (15a)
=
∞∑
j=0
Ω(j)
[
a
b, d
∣∣∣∣∣ q
]
j
3φ2
[
qja, c, e
qjb, qjd
∣∣∣∣∣ q; qjbdace
]
(15b)
and then reformulating the last 3φ2-series via (2a), (2b) as
3φ2
[
qja, c, e
qjb, qjd
∣∣∣∣∣ q; qjbdace
]
=
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 11
1548 W. CHU, W. ZHANG
=
[
qjbd/ae, q2jbd/ce
q2jbd/e, qjbd/ace
∣∣∣∣∣ q
]
∞
∞∑
i=0
1− q2i+2j−1bd/e
1− q2j−1bd/e
×
×(−1)iq(
i
2)
[
q2j−1bd/e, qja, c, qjd/e, qjb/e
q, qjbd/ae, q2jbd/ce, qjb, qjd
∣∣∣∣∣ q
]
i
(
qjbd
ac
)i
=
=
[
bd/ae, bd/ce
bd/e, bd/ace
∣∣∣∣∣ q
]
∞
[
b, d, bd/ace
a, b/e, d/e
∣∣∣∣∣ q
]
j
∞∑
i=0
1− q2i+2j−1bd/e
1− bd/qe
×
×(−1)iq(
i
2)
[
a, b/e, d/e
b, d, bd/ae
∣∣∣∣∣ q
]
i+j
[
c
q
∣∣∣∣∣ q
]
i
[
bd/qe
bd/ce
∣∣∣∣∣ q
]
i+2j
(
qjbd
ac
)i
we get the following double sum expression
Eq(14a) =
∞∑
i,j=0
(−1)i
1− q2i+2j−1bd/e
1− bd/qe
q(
i
2)
[
bd/ace
b/e, d/e
∣∣∣∣∣ q
]
j
Ω(j)×
×
[
c
q
∣∣∣∣∣ q
]
i
[
a, b/e, d/e
b, d, bd/ae
∣∣∣∣∣ q
]
i+j
[
bd/qe
bd/ce
∣∣∣∣∣ q
]
i+2j
(
qjbd
ac
)i
.
This leads to the transformation displayed in Theorem 4 after having changed the sum-
mation indices by n := i+ j and then applied the relation
[
c
q
∣∣∣∣∣ q
]
n−j
[
bd/qe
bd/ce
∣∣∣∣∣ q
]
n+j
=
(q
c
)j [c, bd/qe
q, bd/ce
∣∣∣∣∣ q
]
n
[
q−n, qn−1bd/e
qnbd/ce, q1−n/c
∣∣∣∣∣ q
]
j
.
Theorem 4 is proved.
4.1. Specifying the Ω(j) sequence in Theorem 4 by
Ω(j) = (−1)jq(
j
2)
[
b/e, d/e, q/c2
q, bd/ace
∣∣∣∣∣ q
]
j
(
bd
a
)j
and then evaluating the inner sum with respect to j displayed in (14c) by means of the
q-Pfaff – Saalschütz formula (6), we obtain the following reduction formula.
Proposition 9 (well-posed reduction formula).
∞∑
i,j=0
(−1)jq(
j
2)
[
a
b, d
∣∣∣∣∣ q
]
i+j
[
c, e
q
∣∣∣∣∣ q
]
i
[
b/e, d/e, q/c2
q, bd/ace
∣∣∣∣∣ q
]
j
(
qjbd
ace
)i(
bd
a
)j
=
=
[
bd/ae, bd/ce
bd/e, bd/ace
∣∣∣∣∣ q
]
∞
∞∑
n=0
(−1)nq(
n
2) 1− q2n−1bd/e
1− bd/qe
×
×
[
bd/qe, a, q/c, b/e, d/e
q, b, d, bd/ae, bcd/qe
∣∣∣∣∣ q
]
n
[
bcd/qe
bd/ce
∣∣∣∣∣ q
]
2n
(
bd
ac
)n
.
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 11
WELL-POSED REDUCTION FORMULAE FOR q-KAMPÉ DE FÉRIET FUNCTION 1549
4.2. Instead, specifying the Ω(j) sequence by
Ω(j) =
= (−1)jq(
j
2) 1−q2j−1bd/ce
1− bd/qce
[
α/c, β/c, bd/qce, bd/αβe
q, αβ/c, bd/αe, bd/βe
∣∣∣∣∣ q
]
j
[
b/e, d/e
bd/ace
∣∣∣∣∣ q
]
j
(
bd
a
)j
and then evaluating the inner sum with respect to j displayed in (14c) by means of
Jackson’s q-analogue of Dougall’s 7F6-sum (cf. [5], II-22)
8φ7
[
a, q
√
a, −q
√
a, b, c, d, e, q−n
√
a, −
√
a, qa/b, qa/c, qa/d, qa/e, qn+1a
∣∣∣∣∣ q; q
]
=
=
[
qa, qa/bc, qa/bd, qa/cd
qa/b, qa/c, qa/d, qa/bcd
∣∣∣∣∣ q
]
n
where qn+1a2 = bcde
we get from Theorem 4 another reduction formula.
Proposition 10 (well-posed reduction formula).
∞∑
i,j=0
[
a
b, d
∣∣∣∣∣ q
]
i+j
[
c, e
q
∣∣∣∣∣ q
]
i
[
b/e, d/e
bd/ace
∣∣∣∣∣ q
]
j
(
qjbd
ace
)i(
bd
a
)j
×
×(−1)jq(
j
2) 1− q2j−1bd/ce
1− bd/qce
[
α/c, β/c, bd/qce, bd/αβe
q, αβ/c, bd/αe, bd/βe
∣∣∣∣∣ q
]
j
=
=
[
bd/ae, bd/ce
bd/e, bd/ace
∣∣∣∣∣ q
]
∞
∞∑
n=0
(−1)nq(
n
2) 1− q2n−1bd/e
1− bd/qe
×
×
[
a, α, β, b/e, d/e, bd/qe, bcd/αβe
q, b, d, αβ/c, bd/ae, bd/αe, bd/βe
∣∣∣∣∣ q
]
n
(
bd
ac
)n
.
5. Nonterminating double series Φ1:2;λ
1:1;µ. Applying the two transformations for
3φ2-series (cf. [5], III-9 and III-10), Jia, Wang [7] studied systematically this Φ1:2;λ
1:1;µ
series and found several reduction and summation formulae. Alternatively, we shall
employ (2a), (2b) to show a couple of new transformation theorems for this Φ1:2;λ
1:1;µ series
and deduce from them four very strange reduction formulae, that differ substantially
from those due to Jia, Wang [7].
Theorem 5. For an arbitrary sequence {Ω(j)}, there holds the transformation[
bd/e, bd/ace
bd/ae, bd/ce
∣∣∣∣∣ q
]
∞
∞∑
i,j=0
(
bd
ace
)i [a
b
∣∣∣∣∣ q
]
i+j
[
c, e
q, d
∣∣∣∣∣ q
]
i
Ω(j) = (16a)
=
∞∑
i,j=0
1− q2i+j−1bd/e
1− bd/qe
[
a, b/e, bd/qe
b, bd/ce
∣∣∣∣∣ q
]
i+j
× (16b)
×q(
i
2)
[
c, d/e
q, d, bd/ae
∣∣∣∣∣ q
]
i
(
−bd
ac
)i
Ω(j)
(b/e; q)j
(16c)
provided that both double series displayed above are absolutely convergent.
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 11
1550 W. CHU, W. ZHANG
Proof. Rewrite the double sum in (16a) as
∞∑
i,j=0
(
bd
ace
)i [a
b
∣∣∣∣∣ q
]
i+j
[
c, e
q, d
∣∣∣∣∣ q
]
i
Ω(j) = (17a)
=
∞∑
j=0
Ω(j)
[
a
b
∣∣∣∣∣ q
]
j
3φ2
[
qja, c, e
qjb, d
∣∣∣∣∣ q; bdace
]
. (17b)
According to (2a), (2b), the last 3φ2-series can be reformulated as
3φ2
[
qja, c, e
qjb, d
∣∣∣∣∣ q; bdace
]
=
[
bd/ae, qjbd/ce
qjbd/e, bd/ace
∣∣∣∣∣ q
]
∞
∞∑
i=0
(−1)i
1− q2i+j−1bd/e
1− qj−1bd/e
q(
i
2)×
×
[
qj−1bd/e, qja, c, qjb/e, d/e
q, bd/ae, qjbd/ce, d, qjb
∣∣∣∣∣ q
]
i
(
bd
ac
)i
=
=
[
bd/ae, bd/ce
bd/e, bd/ace
∣∣∣∣∣ q
]
∞
[
b
a, b/e
∣∣∣∣∣ q
]
j
∞∑
i=0
1− q2i+j−1bd/e
1− bd/qe
×
×(−1)iq(
i
2)
[
a, b/e, bd/qe
b, bd/ce
∣∣∣∣∣ q
]
i+j
[
c, d/e
q, d, bd/ae
∣∣∣∣∣ q
]
i
(
bd
ac
)i
.
Substituting this expression into (17a), (17b) and then reordering the factors, we get the
transformation displayed in Theorem 5.
Alternatively, permuting the parameters of 3φ2-series gives
3φ2
[
qja, c, e
qjb, d
∣∣∣∣∣ q; bdace
]
= 3φ2
[
c, e, qja
qjb, d
∣∣∣∣∣ q; bdace
]
.
Applying the formula (2a), (2b) to the 3φ2-series on the right-hand side yields
3φ2
[
c, e, qja
qjb, d
∣∣∣∣∣ q; bdace
]
=
[
bd/ac, bd/ae
bd/a, bd/ace
∣∣∣∣∣ q
]
∞
∞∑
i=0
(−1)i
1− q2i−1bd/a
1− bd/qa
q(
i
2)×
×
[
bd/qa, c, e, b/a, q−jd/a
q, bd/ac, bd/ae, d, qjb
∣∣∣∣∣ q
]
i
(
qjbd
ce
)i
=
=
[
bd/ac, bd/ae
bd/a, bd/ace
∣∣∣∣∣ q
]
∞
∞∑
i=0
1− q2i−1bd/a
1− bd/qa
q2(
i
2)×
×
[
bd/qa, c, e, b/a
q, bd/ac, bd/ae, d
∣∣∣∣∣ q
]
i
[b, qa/d; q]j
(b; q)i+j(qa/d; q)j−i
(
bd2
ace
)i
.
Substituting this expression into (17a), (17b) and then reordering the factors, we obtain
another transformation formula.
Theorem 6. For an arbitrary sequence {Ω(j)}, there holds the transformation[
bd/a, bd/ace
bd/ac, bd/ae
∣∣∣∣∣ q
]
∞
∞∑
i,j=0
(
bd
ace
)i [a
b
∣∣∣∣∣ q
]
i+j
[
c, e
q, d
∣∣∣∣∣ q
]
i
Ω(j) = (18a)
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 11
WELL-POSED REDUCTION FORMULAE FOR q-KAMPÉ DE FÉRIET FUNCTION 1551
=
∞∑
i,j=0
1− q2i−1bd/a
1− bd/qa
[
c, e, b/a, bd/qa
q, d, bd/ac, bd/ae
∣∣∣∣∣ q
]
i
× (18b)
×q2(
i
2)
(
bd2
ace
)i
[a, qa/d; q]j
(b; q)i+j(qa/d; q)j−i
Ω(j) (18c)
provided that both double series displayed above are absolutely convergent.
For the two equations displayed in Theorems 5 and 6, our efforts have failed to
reduce the double sums on the right-hand side. However, we do succeed in figuring out
two instances in which their corresponding left double sums can be expressed in single
ones, that lead us to four remarkable reduction formulae.
5.1. Rewrite the double sum in (16a) or the same (18a) as
∞∑
i=0
[
a, c, e
q, b, d
∣∣∣∣∣ q
]
i
(
bd
ace
)i ∞∑
j=0
[
qia
qib
∣∣∣∣∣ q
]
j
Ω(j). (19)
For the Ω(j) sequence specified by
Ω(j) =
[
β
q
∣∣∣∣∣ q
]
j
(
b
aβ
)j
the sum with respect to j displayed in (19) can be evaluated via (10) as
2φ1
[
β, qia
qib
∣∣∣∣∣ q; b
aβ
]
=
[
b/a, qib/β
qib, b/aβ
∣∣∣∣∣ q
]
∞
.
Then Theorems 5 and 6 under the last specification for the Ω(j) sequence give rise
respectively to the following two reduction formulae.
Proposition 11 (reduction formula).
∞∑
i,j=0
1− q2i+j−1bd/e
1− bd/qe
[
a, b/e, bd/qe
b, bd/ce
∣∣∣∣∣ q
]
i+j
×
×q(
i
2)
[
c, d/e
q, d, bd/ae
∣∣∣∣∣ q
]
i
(
−bd
ac
)i [ β
q, b/e
∣∣∣∣∣ q
]
j
(
b
aβ
)j
=
=
[
b/a, b/β, bd/e, bd/ace
b, b/aβ, bd/ae, bd/ce
∣∣∣∣∣ q
]
∞
3φ2
[
a, c, e
d, b/β
∣∣∣∣∣ q; bdace
]
.
Proposition 12 (reduction formula).
∞∑
i,j=0
1− q2i−1bd/a
1− bd/qa
[
c, e, b/a, bd/qa
q, d, bd/ac, bd/ae
∣∣∣∣∣ q
]
i
×
×q2(
i
2)
(
bd2
ace
)i
[β, a, qa/d; q]j
(q; q)j(b; q)i+j(qa/d; q)j−i
(
b
aβ
)j
=
=
[
b/a, b/β, bd/a, bd/ace
b, b/aβ, bd/ac, bd/ae
∣∣∣∣∣ q
]
∞
3φ2
[
a, c, e
d, b/β
∣∣∣∣∣ q; bdace
]
.
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 11
1552 W. CHU, W. ZHANG
5.2. Alternatively, letting Ω(j) be the sequence
Ω(j) =
(qa/b; q)j
(q; q)j
(
−b
a
)j
and then evaluating the corresponding sum with respect to j displayed in (19) by means
of the q-Kummer sum (cf. [5], II-9)
2φ1
[
qia, qa/b
qib
∣∣∣∣∣ q;−b/a
]
= (−q; q)∞
[q1+ia, qib2/a; q2]∞
[qib,−b/a; q]∞
we derive from Theorems 5 and 6 the following respective reduction formulae.
Proposition 13 (reduction formula).
∞∑
i,j=0
1− q2i+j−1bd/e
1− bd/qe
[
a, b/e, bd/qe
b, bd/ce
∣∣∣∣∣ q
]
i+j
(
−bd
ac
)i
×
×q(
i
2)
[
c, d/e
q, d, bd/ae
∣∣∣∣∣ q
]
i
[
qa/b
q, b/e
∣∣∣∣∣ q
]
j
(
−b
a
)j
=
=
[
−q, a, bd/e, bd/ace
−b/a, b, bd/ae, bd/ce
∣∣∣∣∣ q
]
∞
∞∑
i=0
[
c, e
q, d
∣∣∣∣∣ q
]
i
[
qib2/a
qia
∣∣∣∣∣ q2
]
∞
(
bd
ace
)i
.
Proposition 14 (reduction formula).
∞∑
i,j=0
1− q2i−1bd/a
1− bd/qa
[
c, e, b/a, bd/qa
q, d, bd/ac, bd/ae
∣∣∣∣∣ q
]
i
×
×q2(
i
2)
(
bd2
ace
)i
[a, qa/b, qa/d; q]j
(q; q)j(b; q)i+j(qa/d; q)j−i
(
−b
a
)j
=
=
[
−q, a, bd/a, bd/ace
−b/a, b, bd/ac, bd/ae
∣∣∣∣∣ q
]
∞
∞∑
i=0
[
c, e
q, d
∣∣∣∣∣ q
]
i
[
qib2/a
qia
∣∣∣∣∣ q2
]
∞
(
bd
ace
)i
.
6. Nonterminating double series Φ0:3;λ
2:0;µ. Finally in this section, we are going to
investigate the Φ0:3;λ
2:0;µ series and prove one transformation theorem plus two interesting
reduction formulae.
Theorem 7. For an arbitrary sequence {Ω(j)}, there holds the transformation[
bd/e, bd/ace
bd/ae, bd/ce
∣∣∣∣∣ q
]
∞
∞∑
i,j=0
Ω(j)
[b, d; q]i+j
[
a, c, e
q
∣∣∣∣∣ q
]
i
(
q2jbd
ace
)i
= (20a)
=
∞∑
i,j=0
1− q2i+2j−1bd/e
1− bd/qe
[
bd/qe
bd/ae, bd/ce
∣∣∣∣∣ q
]
i+2j
[
a, c
q
∣∣∣∣∣ q
]
i
× (20b)
×q(
i
2)+2ij
[
b/e, d/e
b, d
∣∣∣∣∣ q
]
i+j
(
−bd
ac
)i
(bd/ace; q)2j
[b/e, d/e; q]j
Ω(j) (20c)
provided that both double series displayed above are absolutely convergent.
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 11
WELL-POSED REDUCTION FORMULAE FOR q-KAMPÉ DE FÉRIET FUNCTION 1553
Proof. Rewrite the double sum in (20a) as
∞∑
i,j=0
Ω(j)
[b, d; q]i+j
[
a, c, e
q
∣∣∣∣∣ q
]
i
(
q2jbd
ace
)i
=
∞∑
j=0
Ω(j)
[b, d; q]j
3φ2
[
a, c, e
qjb, qjd
∣∣∣∣∣ q; q2jbdace
]
.
(21)
According to (2a), (2b), we can reformulate the above 3φ2-series as
3φ2
[
a, c, e
qjb, qjd
∣∣∣∣∣ q; q2jbdace
]
=
[
q2jbd/ae, q2jbd/ce
q2jbd/e, q2jbd/ace
∣∣∣∣∣ q
]
∞
∞∑
i=0
1− q2i+2j−1bd/e
1− q2j−1bd/e
×
×(−1)iq(
i
2)
[
q2j−1bd/e, a, c, qjd/e, qjb/e
q, q2jbd/ae, q2jbd/ce, qjb, qjd
∣∣∣∣∣ q
]
i
(
q2jbd
ac
)i
=
=
[
bd/ae, bd/ce
bd/e, bd/ace
∣∣∣∣∣ q
]
∞
[b, d; q]j(bd/ace; q)2j
[b/e, d/e; q]j
∞∑
i=0
1− q2i+2j−1bd/e
1− bd/qe
×
×(−1)iq(
i
2)+2ij
[
b/e, d/e
b, d
∣∣∣∣∣ q
]
i+j
[
a, c
q
∣∣∣∣∣ q
]
i
[
bd/qe
bd/ae, bd/ce
∣∣∣∣∣ q
]
i+2j
(
bd
ac
)i
.
Substituting this last expression into (21) and then simplifying the resulting equation,
we get the transformation displayed in Theorem 7.
By specifying the Ω(j) sequence and then expressing the double sum (20a) as single
series, we can prove two quite interesting reduction formulae.
6.1. Letting b = −d =
√
α and replacing e by −e, we can reformulate the double
sum in (20a) as
∞∑
i=0
[
a, c, −e
q,
√
α, −
√
α
∣∣∣∣∣ q
]
i
( α
ace
)i ∞∑
j=0
Ω(j)
(q2iα; q2)j
q2ij . (22)
Specifying the Ω(j) sequence in Theorem 7 by
Ω(j) =
[
β, γ
q2
∣∣∣∣∣ q2
]
j
(
α
βγ
)j
and then evaluating the inner sum with respect to j displayed in (22) by means of the
q-Gauss sum (10), we get the following reduction formula.
Proposition 15 (reduction formula).
∞∑
i,j=0
1− q2i+2j−1α/e
1− α/qe
[
α/qe
α/ae, α/ce
∣∣∣∣∣ q
]
i+2j
[
α/e2
α
∣∣∣∣∣ q2
]
i+j
×
×q(
i
2)
[
a, c
q
∣∣∣∣∣ q
]
i
(
q2jα
ac
)i
(α/ace; q)2j
[
β, γ
q2, α/e2
∣∣∣∣∣ q2
]
j
(
α
βγ
)j
=
=
[
α/β, α/γ
α, α/βγ
∣∣∣∣∣ q2
]
∞
[
α/e, α/ace
α/ae, α/ce
∣∣∣∣∣ q
]
∞
∞∑
i=0
[a, c,−e; q]i(α/βγ; q2)i
(q; q)i[α/β, α/γ; q2]i
( α
ace
)i
.
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 11
1554 W. CHU, W. ZHANG
6.2. Instead, specifying the Ω(j) sequence in Theorem 7 by
Ω(j) =
[
β, q2/β
q2, −q2
∣∣∣∣∣ q2
]
j
(qj−1α)j
and then evaluating the inner sum with respect to j displayed in (22) by means of the
q-analogue of Bailey’s 2F1( 1
2 ) sum (11), we obtain another reduction formula.
Proposition 16 (reduction formula).
∞∑
i,j=0
1− q2i+2j−1α/e
1− α/qe
[
α/qe
α/ae, α/ce
∣∣∣∣∣ q
]
i+2j
[
α/e2
α
∣∣∣∣∣ q2
]
i+j
[
a, c
q
∣∣∣∣∣ q
]
i
q(
i
2)×
×
(
q2jα
ac
)i [β, q2/β
α/e2
∣∣∣∣∣ q2
]
j
(α/ace; q)2j
(q4; q4)j
(qj−1α)j =
=
[
α/β
α
∣∣∣∣∣ q2
]
∞
[
α/e, α/ace
α/ae, α/ce
∣∣∣∣∣ q
]
∞
∞∑
i=0
[a, c,−e; q]i
(q; q)i(α/β; q2)i
[
q2iαβ
q2iα/β
∣∣∣∣∣ q4
]
∞
( α
ace
)i
.
1. Bailey W. N. Generalized hypergeometric series. – Cambridge: Cambridge Univ. Press, 1935.
2. Chu W., Jia C. Bivariate classical and q-series transformations // Port. math. – 2008. – 65, № 2. –
P. 243 – 256.
3. Chu W., Jia C. Transformation and reduction formulae for double q-Clausen hypergeometric series //
Math. Methods Appl. Sci. – 2008. – 31, № 1. – P. 1 – 17.
4. Chu W., Srivastava H. M. Ordinary and basic bivariate hypergeometric transformations associated with
the Appell and Kampé de Fériet functions // J. Comput. and Appl. Math. – 2003. – 156, № 2. – P. 355 – 370.
5. Gasper G., Rahman M. Basic hypergeometric series. – 2nd ed. – Cambridge: Cambridge Univ. Press,
2004.
6. Van der Jeugt J. Transformation formula for a double Clausenian hypergeometric series, its q-analogue,
and its invariance group // J. Comput. and Appl. Math. – 2002. – 139, № 1. – P. 65 – 73.
7. Jia C., Wang T. Transformation and reduction formulae for double q-Clausen series of type Φ1:2;λ
1:1;µ // J.
Math. Anal. and Appl. – 2007. – 328, № 1. – P. 609 – 624.
8. Jia C., Wang T. Reduction and transformation formulae for bivariate basic hypergeometric series // Ibid.
№ 2. – P. 1152 – 1160.
9. Singh S. P. Certain transformation formulae involving basic hypergeometric functions // J. Math. Phys.
Sci. – 1994. – 28, № 4. – P. 189 – 195.
10. Srivastava H. M., Karlsson P. W. Multiple Gaussian hypergeometric series. – New York etc.: John Wiley
and Sons, 1985.
Received 19.02.10
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 11
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