On a New Summation Formula for

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Volume 2011, Article ID 132081,7pages doi:10.1155/2011/132081

Research Article

On a New Summation Formula for

2

ψ

2

Basic Bilateral Hypergeometric Series

and Its Applications

D. D. Somashekara, K. Narasimha Murthy, and S. L. Shalini

Department of Studies in Mathematics, University of Mysore, Manasagangotri, Mysore 570006, India

Correspondence should be addressed to K. Narasimha Murthy,[email protected]

Received 7 December 2010; Accepted 24 January 2011

Academic Editor: J. Dydak

Copyrightq2011 D. D. Somashekara et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We have obtained a new summation formula for2ψ2 bilateral basic hypergeometric series by the method of parameter augmentation and demonstrated its various uses leading to some development of etafunctions,q-gamma, andq-beta function identities.

1. Introduction

The summation formulae for hypergeometric series form a very interesting and useful component of the theory ofbasichypergeometric series. Theq-binomial theorem of Cauchy

1is perhaps the first identity in the class of the summation formulae, which can be stated as

∞

k0

a_k

q_kzk

az_∞

z_∞ , |z|<1, q<1, 1.1

where

a_∞:a;q_∞:

∞

k0

1−aqk,

a_k:a;q_k: a∞ aqk ∞

, k is an integer.

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For more details about theq-binomial theorem and about the identities which fall in this sequel, one may refer to the book by Gasper and Rahman2. Another famous identity in the sequel is the Ramanujan’s 1ψ1summation formula3

∞

k−∞

a_k b_kzk

az_∞q_∞q/az_∞b/a_∞ z_∞b_∞b/az_∞q/a_∞ ,

b_a<|z|<1, q<1. 1.3

There are a number of proofs of the1ψ1 summation formula1.3in the literature. For more

details, one refers to the book by Berndt4and a recent paper of Johnson5.

In this paper, we derive a new summation formula for2ψ2basic bilateral

hypergeomet-ric series using the1ψ1summation formula1.3by the method of parameter augmentation.

We then use the formula to derive theq-analogue of Gauss summation formula and to obtain a number of etafunction, q-gamma, and q-beta function identities, which complement the works of Bhargava and Somashekara6, Bhargava et al.7, Somashekara and Mamta8, Srivastava9, and Bhargava and Adiga10.

First, we recall thatq-diﬀerence operator and theq-shift operator are defined by

Dqfa fa−f

aq

a , ζ

fafaq, 1.4

respectively. In11, Chen and Liu have constructed an operatorθas

θζ−1_D

q, 1.5

and thereby they defined the operatorEbθas

Ebθ ∞

k0

bθk_qkk−1/2

q;q_k . 1.6

Then, we have the following operator identities12, Theorem 1:

Ebθ at;q_∞at, bt;q_∞,

Ebθ as, at;q_∞

as, at, bs, bt;q_∞

abst/q;q_∞ ,

abst_q <1.

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Further, the Dedekind etafunction is defined by

ητ:eπiτ/12

∞

k1

1−e2πikτ:q1/24q;q_∞, 1.8

whereqe2πiτ, and Imτ>0.

Jackson13defined theq-analogue of the gamma function by

Γqx

q;q_∞

qx_;_q ∞

1−q1−x, 0< q <1. 1.9

In his paper on the q-gamma andq-beta function, Askey14has obtainedq-analogues of several classical results about the gamma function. Further, he has given the definition for q-beta function as

Bqx, y1−q ∞

n0

qnx

qn 1 ∞

qx y ∞

. 1.10

In fact, he has shown that

Bqx, y ΓqxΓq

y

Γqx y . 1.11

InSection 2, we prove our main result. In Section 3, we deduce the well-known q-analogue of the Gauss summation formula and some etafunction, q-gamma, and q-beta function identities.

2. Main Result

Theorem 2.1. If 0<|z|<1, |q|<1, then

∞

k−∞

a_kbc/azq_k

b_kc_k zk

az_∞q_∞q/az_∞b/a_∞c/a_∞bc/azq_∞

z_∞b_∞c_∞b/az_∞c/az_∞q/a_∞ . 2.1

Proof. Ramanujan’s1ψ1summation formula1.3can be written as

∞

k0

a_k b_kzk

∞

k1

q/b_k

q/a_k

_b

az

_k

az∞

q_∞q/az_∞b/a_∞

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This is the same as

∞

k0

a_kzk_bqk ∞

_b

az

∞

k1

−1kqkk 1/2

q/a_k

1 az

k

bq−k ∞

_b

az

∞

az∞

q_∞q/az_∞

z_∞q/a_∞

_b

a

∞

.

2.3

On applyingEcθto both sides with respect tob, we obtain

∞

k0

a_kzk

bqk

∞b/az ∞cqk∞c/az∞

bcqk_/azq ∞

∞

k1

−1kqkk 1/2

q/a_k

1 az

_k_bq−k

∞b/az ∞cq−k∞c/az∞

bcq−k_/azq ∞

az∞

q_∞q/az_∞

z_∞q/a_∞

_b

a

∞

_c

a

∞

.

2.4

Multiplying2.4throughout by{bc/azq_∞/b_∞c_∞}, we obtain

∞

k0

a_kbc/azq_k

b_kc_k zk

∞

k1

a_−kbc/azq_−k

b_−kc_−k z−k

az∞

q_∞q/az_∞b/a_∞c/a_∞bc/azq_∞

z_∞b_∞c_∞b/az_∞c/az_∞q/a_∞ ,

2.5

which yields2.1.

3. Some Applications of the Main Identity

The following identity is the well-knownq-analogue of the Gauss summation formula.

Corollary 3.1see15. If|q|<1,|γ/αβ|<1, then

∞

k0

α_kβ_k

q_kγ_k

_γ

αβ

_k

γ/α_∞γ/β_∞

γ_∞γ/αβ_∞ . 3.1

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Corollary 3.2. If|q|<1, then

∞

k−∞

q2_;_q4 k

1−q4k 1q3_;q4 k

qk_q−1/8 η72τ

η3_τ_η2_4τ, 3.2

∞

k−∞

−q2_;_q2

k

1 q2k 12−q;q2 k

qk_2q−3/4η64τ

η3_2τ, 3.3

∞

k−∞

−q;q2

k

−q2_;_q2

k

1−q2k 12q;q22 n

qk_2q−3/4η84τ

η7_2τ, 3.4

∞

k−∞

1 q−q;q2 k

−q2_;_q2

k

1−q2k 2q2_;_q22 k

qk η2τ

η2_τ, 3.5

∞

k−∞

q2_;_q6 k

1−q6k 1q5_;q6 k

q3k_q−1/4η42τ

η2_τ , 3.6

∞

k−∞

1 q22₋_q2_;_q2 k−1

−q2_;_q2

k 1

1−q2k 22q2_;_q22 k

qk η24τ

η4_2τ. 3.7

Proof. Puttingazq1/4,bq3/4,cq5/4, and then changingqtoq4_in_2.1_{, we obtain}

∞

k−∞

q;q4 k

q2_;_q4 k

q;q4 k 1

q3_;_q4 k

qk

q2_;_q44 ∞

q4_;_q42 ∞

q;q43 ∞

q3_;_q43 ∞

. 3.8

Simplifying the right hand side and then using1.8, we obtain3.2.

Similarly, puttinga −q1/2,zq1/2,b−q3/2,c −q3/2, and then changingqtoq2_,

we obtain3.3. Puttinga−q1/2,zq1/2,bcq3/2, and then changingqtoq2_{, we obtain}

3.4. Puttinga −q1/2,z q1/2,b q,c q2_{, and then changing}_q_to_q2_{, we obtain}_3.5_.

Puttingaq1/6,zq1/2,bq7/6,cq5/6, and then changingqtoq6_{, we obtain}_3.6_{. Finally,}

puttinga−1,zq,bcq2, and then changingqtoq2, we obtain3.7.

Corollary 3.3. If 0< q <1, 0< x,y <1, and 0< x y <1, then

B3 q

x, y Γq

1−x yΓqx−y1−q2

1−qy2

∞ k−∞ q1−x k

qx y k

q1 y2 k

qky_. _3.9

Proof. Puttingaq1−x,zqy, andb c q1 yin2.1, we get

∞ k−∞ q1−x k

qx y k

q1 y2 k

qky

q1−x y

∞

q_∞qx−y ∞

qy ∞

q1 y ∞

qx y ∞

qx ∞

3

. 3.10

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B2

qx, y

Γqy−xΓqx−y 1Γqx y−1

ΓqyΓq1 x

∞

k−∞

q−x k

qx y−1

k

qy2 k

qky_. _3.11

Proof. Puttinga q−x,z b c qy, in2.1, and then using 1.9,1.10, and1.11, we

obtain3.11.

Corollary 3.5. If 0< x,y <1, and 0< x y <1, then

B3_{x, y} Γ

1−x yΓx−y y2

⎡

⎣∞

k0

1−x_kx y_k

1 y2_k

∞

k1

−y2_k

x_k1−x−y_k

⎤

⎦_. 3.12

Proof. Lettingq → 1 in3.9, we obtain3.12.

Corollary 3.6. If 0< x,y <1, and 1< x y <2, then

B2_{x, y} Γ

y−xΓx−y 1Γx y−1

ΓyΓ1 x

× ⎡

⎣∞

k0

−x_kx y−1_k

y2 k

∞

k1

1−y2_k

1 x_k2−x−y_k

⎤

⎦_.

3.13

Proof. Lettingq → 1 in3.11, we obtain3.13.

Bqx, y ΓqyΓq1−y ∞

k0

q1−x−y

k

qy k

q2 k

qkx_. _3.14

Proof. Puttingaq1−x−y,zqx, andbcqin2.1, we obtain3.14.

Acknowledgment

The authors would like to thank the referees for their valuable suggestions which consi-derably improved the quality of the paper.

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