On Certain Transformation Formulas Involving Basic Hypergeometric Series
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RESEARCH ARTICLE
On Certain Transformation Formulas Involving Basic Hypergeometric Series Satya Prakash Singh1 • Vijay Yadav2
Received: 5 January 2018 / Revised: 4 April 2019 / Accepted: 8 April 2019 Ó The National Academy of Sciences, India 2019
Abstract In this paper, making use of Bailey’s transform and certain summations of basic hypergeometric series, we have established interesting transformation formulae involving basic hypergeometric series. Keywords Bailey transform Summation formula Transformation formula Basic hypergeometric series Mathematics Subject Classification Primary 33D15 11P83
1 Introduction, Notations and Definitions For real or complex q, jqj\1, let 1 Y ð1 kqi Þ ðk; qÞl ¼ ð1 kqlþi Þ i¼0
ð1Þ
for arbitrary k and l, so that ðk; qÞn ¼
1;
n¼0
ð1 kÞð1 kqÞ ð1 kqn1 Þ;
n ¼ 1; 2; 3; . . .
ðk; qÞ1 ¼
ð1 kqi Þ:
ð3Þ
i¼0
Define as usual, a generalized q-hypergeometric function by [Srivastava and Karlsson [1]; (272) p. 347], see also Slater [2], a1 ; a2 ; . . .; ar ; q; z U r s b1 ; b2 ; . . .; bs ; qk ð4Þ 1 X ða1 ; a2 ; . . .; ar ; qÞn zn qknðn1Þ=2 ¼ ; ðq; b1 ; b2 ; . . .; bs ; qÞn n¼0 where ða1 ; a2 ; a3 ; . . .; ar ; qÞn ¼ ða1 ; qÞn ða2 ; qÞn . . .ðar ; qÞn : Series in Eq. (4) converges for jqj\1; jzj\1 if k [ 0 and if k ¼ 0; then, for r ¼ s þ 1, it converges in the unit circle jzj\1 provided no denominator parameter is of the form qk for k positive integer. Bailey established a simple, remarkable and extremely useful transform, known as Bailey’s transform, which states, If n X bn ¼ ar unr vnþr ð5Þ
ð2Þ and
1 Y
r¼0
and cn ¼
1 X
drþn ur vrþ2n ;
ð6Þ
r¼0
& Satya Prakash Singh [email protected] Vijay Yadav [email protected] 1
Department of Mathematics, T.D.P.G. College, Jaunpur, Uttar Pradesh 222002, India
2
Department of Mathematics and Statistics, S.P.D.T. College, Andheri (East), Mumbai 400059, India
where ur ; vr ; ar and dr are any functions of r alone and the series cn exists, then 1 1 X X an c n ¼ bn dn ; ð7Þ n¼0
n¼0
provided both series of Eq. (7) converge.
123
S. P. Singh, V. Yadav 1r 2 2
r
form takes the following form: If n q 2n X 1 ar vrþn ð1Þr ðqn ; qÞr q2r ðq; qÞn r¼0 1 2
bn ¼
ð8Þ
a1=3 ; xa1=3 ; x2 a1=3 ; aq1þn ; qn ; q; q pffiffiffi pffiffiffi pffiffiffiffiffi pffiffiffiffiffi 5 U4 q a; a; aq; aq pffiffiffi ð aÞnm ðq; qÞn ðaq3 ; q3 Þm ¼ : ðaq; qÞn ðq3 ; q3 Þm
# ð15Þ
[Verma and Jain [3]; (4.5)p. 1036] # pffiffiffi a1=3 ; xa1=3 ; x2 a1=3 ; q a; aq1þn ; qn ; q; q pffiffiffi pffiffiffi pffiffiffiffiffi pffiffiffiffiffi 2 pffiffiffi 6 U5 a; a; aq; aq; q a pffiffiffi pffiffiffi pffiffiffi ðq; qÞn ð a; qÞn ðaq3 ; q3 Þm ðq6 a; q3 Þm ð aÞnm pffiffiffi pffiffiffi ¼ : ðaq; qÞn ðq2 a; qÞn ðq3 ; q3 Þm ð a; q3 Þm "
and 1
c n ¼ q 2n
1 2 1 1 X q 2 r þ 2r
r¼0
ðq; qÞr
n¼0
ð9Þ
vrþ2n
then under suitable convergence conditions, 1 1 X X an c n ¼ bn dn :
ð16Þ ð10Þ
n¼0
We make use of this new form of Bailey’s transform given in Eqs. (8)–(10) in our analysis. The following summation formulae are used in the next section. n 2 2 1þn q ; x y q ; x; xq; q; q 4 U3 xyq; xyq; x2 q ð11Þ xn ðq; qÞn ðx2 q2 ; q2 Þm ðy2 q2 ; q2 Þm ;
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