On a New Summation Formula for _𝟐𝜓_𝟐 Basic Bilateral Hypergeometric Series and Its Applications

Abstract

We have obtained a new summation formula for ₂𝜓₂ bilateral basic hypergeometric series by the method of parameter augmentation and demonstrated its various uses leading to some development of etafunctions, 𝑞-gamma, and 𝑞-beta function identities.

1. Introduction

The summation formulae for hypergeometric series form a very interesting and useful component of the theory of (basic) hypergeometric series. The 𝑞-binomial theorem of Cauchy [1] is perhaps the first identity in the class of the summation formulae, which can be stated as ^∞_𝑘=0(𝑎)_𝑘𝑞_𝑘𝑧^𝑘=(𝑎𝑧)_∞(𝑧)_∞,|𝑧|<1,𝑞<1, (1.1) where(𝑎)_∞∶=𝑎;𝑞_∞∶=^∞_𝑘=01−𝑎𝑞^𝑘,(𝑎)_𝑘∶=𝑎;𝑞_𝑘(𝑎)∶=^∞𝑎𝑞^𝑘_∞,𝑘isaninteger. (1.2) For more details about the 𝑞-binomial theorem and about the identities which fall in this sequel, one may refer to the book by Gasper and Rahman [2]. Another famous identity in the sequel is the Ramanujan's ₁𝜓₁ summation formula [3]^∞_𝑘=−∞(𝑎)_𝑘(𝑏)_𝑘𝑧^𝑘=(𝑎𝑧)_∞𝑞_∞𝑞/𝑎𝑧_∞(𝑏/𝑎)_∞(𝑧)_∞(𝑏)_∞(𝑏/𝑎𝑧)_∞𝑞/𝑎_∞,𝑏𝑎<|𝑧|<1,𝑞<1. (1.3) There are a number of proofs of the₁𝜓₁ summation formula (1.3) in the literature. For more details, one refers to the book by Berndt [4] and a recent paper of Johnson [5].

In this paper, we derive a new summation formula for₂𝜓₂ basic bilateral hypergeometric series using the₁𝜓₁ summation formula (1.3) by the method of parameter augmentation. We then use the formula to derive the 𝑞-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 Somashekara [6], Bhargava et al. [7], Somashekara and Mamta [8], Srivastava [9], and Bhargava and Adiga [10].

First, we recall that 𝑞-difference operator and the 𝑞-shift operator are defined by 𝐷_𝑞𝑓(𝑎)=𝑓(𝑎)−𝑓𝑎𝑞𝑎,𝜁𝑓(𝑎)=𝑓𝑎𝑞, (1.4) respectively. In [11], Chen and Liu have constructed an operator 𝜃 as 𝜃=𝜁⁻¹𝐷_𝑞, (1.5) and thereby they defined the operator 𝐸(𝑏𝜃) as𝐸(𝑏𝜃)=^∞_𝑘=0(𝑏𝜃)^𝑘𝑞^{𝑘(𝑘−1)/2}𝑞;𝑞_𝑘. (1.6) Then, we have the following operator identities [12, Theorem 1]:𝐸(𝑏𝜃)𝑎𝑡;𝑞_∞=𝑎𝑡,𝑏𝑡;𝑞_∞,𝐸(𝑏𝜃)𝑎𝑠,𝑎𝑡;𝑞_∞=𝑎𝑠,𝑎𝑡,𝑏𝑠,𝑏𝑡;𝑞_∞𝑎𝑏𝑠𝑡/𝑞;𝑞_∞,𝑎𝑏𝑠𝑡𝑞<1. (1.7)

Further, the Dedekind etafunction is defined by𝜂(𝜏)∶=𝑒^{∞𝜋𝑖𝜏/12}_𝑘=11−𝑒^{2𝜋𝑖𝑘𝜏}∶=𝑞^1/24𝑞;𝑞_∞, (1.8) where 𝑞=𝑒^{2𝜋𝑖𝜏}, and Im (𝜏)>0.

Jackson [13] defined the 𝑞-analogue of the gamma function byΓ_𝑞(𝑥)=𝑞;𝑞_∞𝑞^𝑥;𝑞_∞1−𝑞^1−𝑥,0<𝑞<1. (1.9) In his paper on the 𝑞-gamma and 𝑞-beta function, Askey [14] has obtained 𝑞-analogues of several classical results about the gamma function. Further, he has given the definition for 𝑞-beta function as𝐵_𝑞𝑥,𝑦=1−𝑞^∞_𝑛=0𝑞^𝑛𝑥𝑞^𝑛+1_∞𝑞^𝑥+𝑦_∞. (1.10) In fact, he has shown that𝐵_𝑞=Γ𝑥,𝑦^𝑞(𝑥)Γ_𝑞𝑦Γ_𝑞.𝑥+𝑦 (1.11)

In Section 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)_𝑘(𝑐)_𝑘𝑧^𝑘=(𝑎𝑧)_∞𝑞_∞𝑞/𝑎𝑧_∞(𝑏/𝑎)_∞(𝑐/𝑎)_∞𝑏𝑐/𝑎𝑧𝑞_∞(𝑧)_∞(𝑏)_∞(𝑐)_∞(𝑏/𝑎𝑧)_∞(𝑐/𝑎𝑧)_∞𝑞/𝑎_∞. (2.1)

Proof. Ramanujan's₁𝜓₁ summation formula (1.3) can be written as ^∞_𝑘=0(𝑎)_𝑘(𝑏)_𝑘𝑧^𝑘+^∞_𝑘=1𝑞/𝑏_𝑘𝑞/𝑎_𝑘𝑏𝑎𝑧^𝑘=(𝑎𝑧)_∞𝑞_∞𝑞/𝑎𝑧_∞(𝑏/𝑎)_∞(𝑧)_∞(𝑏)_∞(𝑏/𝑎𝑧)_∞𝑞/𝑎_∞. (2.2) This is the same as ^∞_𝑘=0(𝑎)_𝑘𝑧^𝑘𝑏𝑞^𝑘_∞𝑏𝑎𝑧_∞+^∞_𝑘=1(−1)^𝑘𝑞^{𝑘(𝑘+1)/2}𝑞/𝑎_𝑘1𝑎𝑧^𝑘𝑏𝑞^−𝑘_∞𝑏𝑎𝑧_∞=(𝑎𝑧)_∞𝑞_∞𝑞/𝑎𝑧_∞(𝑧)_∞𝑞/𝑎_∞𝑏𝑎_∞. (2.3) On applying 𝐸(𝑐𝜃) to both sides with respect to 𝑏, we obtain ^∞_𝑘=0(𝑎)_𝑘𝑧^𝑘𝑏𝑞^𝑘_∞(𝑏/𝑎𝑧)_∞𝑐𝑞^𝑘_∞(𝑐/𝑎𝑧)_∞𝑏𝑐𝑞^𝑘/𝑎𝑧𝑞_∞+^∞_𝑘=1(−1)^𝑘𝑞^{𝑘(𝑘+1)/2}𝑞/𝑎_𝑘1𝑎𝑧^𝑘𝑏𝑞^−𝑘_∞(𝑏/𝑎𝑧)_∞𝑐𝑞^−𝑘_∞(𝑐/𝑎𝑧)_∞𝑏𝑐𝑞^−𝑘/𝑎𝑧𝑞_∞=(𝑎𝑧)_∞𝑞_∞𝑞/𝑎𝑧_∞(𝑧)_∞𝑞/𝑎_∞𝑏𝑎_∞𝑐𝑎_∞. (2.4) Multiplying (2.4) throughout by {(𝑏𝑐/𝑎𝑧𝑞)_∞/(𝑏)_∞(𝑐)_∞}, we obtain ^∞_𝑘=0(𝑎)_𝑘𝑏𝑐/𝑎𝑧𝑞_𝑘(𝑏)_𝑘(𝑐)_𝑘𝑧^𝑘+^∞_𝑘=1(𝑎)_−𝑘𝑏𝑐/𝑎𝑧𝑞_−𝑘(𝑏)_−𝑘(𝑐)_−𝑘𝑧^−𝑘=(𝑎𝑧)_∞𝑞_∞𝑞/𝑎𝑧_∞(𝑏/𝑎)_∞(𝑐/𝑎)_∞𝑏𝑐/𝑎𝑧𝑞_∞(𝑧)_∞(𝑏)_∞(𝑐)_∞(𝑏/𝑎𝑧)_∞(𝑐/𝑎𝑧)_∞𝑞/𝑎_∞, (2.5) which yields (2.1).

3. Some Applications of the Main Identity

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

Corollary 3.1 (see [15]). If |𝑞|<1, |𝛾/𝛼𝛽|<1, then ^∞_𝑘=0(𝛼)_𝑘𝛽_𝑘𝑞_𝑘𝛾_𝑘𝛾𝛼𝛽^𝑘=𝛾/𝛼_∞𝛾/𝛽_∞𝛾_∞𝛾/𝛼𝛽_∞. (3.1)

Proof. Putting 𝑎=𝛼, 𝑏=𝑞, 𝑐=𝛾, and 𝑧=𝛾/𝛼𝛽 in (2.1), we obtain (3.1).

Corollary 3.2. If |𝑞|<1, then ^∞_𝑘=−∞𝑞²;𝑞⁴_𝑘1−𝑞^4𝑘+1𝑞³;𝑞⁴_𝑘𝑞^𝑘=𝑞^−1/8𝜂⁷(2𝜏)𝜂³(𝜏)𝜂²(),4𝜏_{∞𝑘=−∞}−𝑞²;𝑞²_𝑘1+𝑞^2𝑘+12−𝑞;𝑞²_𝑘𝑞^𝑘=2𝑞^−3/4𝜂⁶(4𝜏)𝜂³(),2𝜏_{∞𝑘=−∞}−𝑞;𝑞²_𝑘−𝑞²;𝑞²_𝑘1−𝑞^2𝑘+12𝑞;𝑞²²_𝑛𝑞^𝑘=2𝑞^−3/4𝜂⁸(4𝜏)𝜂⁷(),2𝜏_{∞𝑘=−∞}1+𝑞−𝑞;𝑞²_𝑘−𝑞²;𝑞²_𝑘1−𝑞^2𝑘+2𝑞²;𝑞²²_𝑘𝑞^𝑘=𝜂(2𝜏)𝜂²(𝜏)_{∞𝑘=−∞},𝑞²;𝑞⁶_𝑘1−𝑞^6𝑘+1𝑞⁵;𝑞⁶_𝑘𝑞^3𝑘=𝑞^−1/4𝜂⁴(2𝜏)𝜂²(𝜏)_{∞𝑘=−∞},1+𝑞²²−𝑞²;𝑞²_𝑘−1−𝑞²;𝑞²_𝑘+11−𝑞^2𝑘+22𝑞²;𝑞²²_𝑘𝑞^𝑘=𝜂²(4𝜏)𝜂⁴().2𝜏 (3.2)(3.3)(3.4)(3.5)(3.6)(3.7)

Proof. Putting 𝑎=𝑧=𝑞^1/4, 𝑏=𝑞^3/4, 𝑐=𝑞^5/4, and then changing 𝑞 to 𝑞⁴ in (2.1), we obtain ^∞_𝑘=−∞𝑞;𝑞⁴_𝑘𝑞²;𝑞⁴_𝑘𝑞;𝑞⁴_𝑘+1𝑞³;𝑞⁴_𝑘𝑞^𝑘=𝑞²;𝑞⁴⁴_∞𝑞⁴;𝑞⁴²_∞𝑞;𝑞⁴³_∞𝑞³;𝑞⁴³_∞. (3.8) Simplifying the right hand side and then using (1.8), we obtain (3.2).
Similarly, putting 𝑎=−𝑞^1/2, 𝑧=𝑞^1/2, 𝑏=−𝑞^3/2, 𝑐=−𝑞^3/2, and then changing 𝑞 to 𝑞², we obtain (3.3). Putting 𝑎=−𝑞^1/2,   𝑧=𝑞^1/2, 𝑏=𝑐=𝑞^3/2, and then changing 𝑞 to 𝑞², we obtain (3.4). Putting 𝑎=−𝑞^1/2,  𝑧=𝑞^1/2, 𝑏=𝑞, 𝑐=𝑞², and then changing 𝑞 to 𝑞², we obtain (3.5). Putting 𝑎=𝑞^1/6,   𝑧=𝑞^1/2, 𝑏=𝑞^7/6, 𝑐=𝑞^5/6, and then changing 𝑞 to 𝑞⁶, we obtain (3.6). Finally, putting 𝑎=−1, 𝑧=𝑞, 𝑏=𝑐=𝑞², and then changing 𝑞 to 𝑞², we obtain (3.7).

Corollary 3.3. If 0<𝑞<1, 0<𝑥, 𝑦<1, and 0<𝑥+𝑦<1, then 𝐵³_𝑞=Γ𝑥,𝑦^𝑞1−𝑥+𝑦Γ_𝑞𝑥−𝑦1−𝑞²1−𝑞^𝑦2∞_𝑘=−∞𝑞^1−𝑥_𝑘𝑞^𝑥+𝑦_𝑘𝑞^1+𝑦2_𝑘𝑞^𝑘𝑦. (3.9)

Proof. Putting 𝑎=𝑞^1−𝑥, 𝑧=𝑞^𝑦, and 𝑏=𝑐=𝑞^1+𝑦 in (2.1), we get ^∞_𝑘=−∞𝑞^1−𝑥_𝑘𝑞^𝑥+𝑦_𝑘𝑞^1+𝑦2_𝑘𝑞^𝑘𝑦=𝑞^{1−𝑥+𝑦}_∞𝑞_∞𝑞^𝑥−𝑦_∞𝑞^𝑦_∞𝑞^1+𝑦_∞𝑞^1+𝑦_∞𝑞^𝑥+𝑦_∞𝑞^𝑥_∞³. (3.10) On using (1.9), (1.10), and (1.11), we obtain (3.9).

Corollary 3.4. If 0<𝑞<1, 0<𝑥, 𝑦<1, and 1<𝑥+𝑦<2, then 𝐵²_𝑞=Γ𝑥,𝑦^𝑞𝑦−𝑥Γ_𝑞𝑥−𝑦+1Γ_𝑞𝑥+𝑦−1Γ_𝑞𝑦Γ_𝑞(1+𝑥)^∞_𝑘=−∞𝑞^−𝑥_𝑘𝑞^{𝑥+𝑦−1}_𝑘𝑞^𝑦2_𝑘𝑞^𝑘𝑦. (3.11)

Proof. Putting 𝑎=𝑞^−𝑥, 𝑧=𝑏=𝑐=𝑞^𝑦, in (2.1), and then using (1.9), (1.10), and (1.11), we obtain (3.11).

Corollary 3.5. If 0<𝑥, 𝑦<1, and 0<𝑥+𝑦<1, then 𝐵³=ΓΓ𝑥,𝑦1−𝑥+𝑦𝑥−𝑦𝑦^2∞_𝑘=0(1−𝑥)_𝑘𝑥+𝑦_𝑘1+𝑦²_𝑘+^∞_𝑘=1−𝑦²_𝑘(𝑥)_𝑘1−𝑥−𝑦_𝑘. (3.12)

Proof. Letting 𝑞→1 in (3.9), we obtain (3.12).

Corollary 3.6. If 0<𝑥, 𝑦<1, and 1<𝑥+𝑦<2, then 𝐵²=ΓΓΓ𝑥,𝑦𝑦−𝑥𝑥−𝑦+1𝑥+𝑦−1Γ𝑦Γ()×1+𝑥_∞𝑘=0(−𝑥)_𝑘𝑥+𝑦−1_𝑘𝑦²_𝑘+^∞_𝑘=11−𝑦²_𝑘(1+𝑥)_𝑘2−𝑥−𝑦_𝑘. (3.13)

Proof. Letting 𝑞→ 1 in (3.11), we obtain (3.13).

Corollary 3.7. If 0<𝑞<1, 0<𝑥, 𝑦<1, and 0<𝑥+𝑦<1, then 𝐵_𝑞𝑥,𝑦=Γ_𝑞𝑦Γ_𝑞1−𝑦^∞_𝑘=0𝑞^{1−𝑥−𝑦}_𝑘𝑞^𝑦_𝑘𝑞²_𝑘𝑞^𝑘𝑥. (3.14)

Proof. Putting 𝑎=𝑞^{1−𝑥−𝑦}, 𝑧=𝑞^𝑥, and 𝑏=𝑐=𝑞 in (2.1), we obtain (3.14).

Acknowledgment

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