Several transformation formulas involving bilateral basic hypergeometric series
Chuanan Wei, Tong Yu
TL;DR
This paper proves three new transformation formulas for bilateral basic hypergeometric series using analytic continuation, including a result equivalent to Jouhet's involving $_8 heta_8$ and $_8 ext{phi}_7$ series.
Contribution
It introduces three novel transformation formulas for bilateral basic hypergeometric series, expanding the theoretical framework of hypergeometric series transformations.
Findings
Proved three new transformation formulas for bilateral basic hypergeometric series
One formula is equivalent to Jouhet's result involving $_8 heta_8$ and $_8 ext{phi}_7$ series
Enhanced understanding of hypergeometric series transformations
Abstract
In terms of the analytic continuation method, we prove three transformation formulas involving bilateral basic hypergeometric series. One of them is equivalent to Jouhet's result involving two series and two series.
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Taxonomy
TopicsMathematical functions and polynomials · Advanced Mathematical Identities · Polynomial and algebraic computation
††2010 Mathematics Subject Classification: Primary 05A30 and Secondary 33D15.
Several transformation formulas involving bilateral basic hypergeometric series
Chuanan Wei, Tong Yu∗
School of Biomedical Information and Engineering
Hainan Medical University, Haikou 571199, China
Abstract.
In terms of the analytic continuation method, we prove three transformation formulas involving bilateral basic hypergeometric series. One of them is equivalent to Jouhet’s result involving two series and two series.
Key words and phrases:
The analytic continuation method; Bilateral basic hypergeometric series; Unilateral basic hypergeometric series
Email addresses: [email protected] (C. Wei), [email protected] (T. Yu).
The corresponding author∗.
1. Introduction
For any integer and complex numbers , with , define the -shifted factorial to be
[TABLE]
For convenience, we shall also adopt the following notations:
[TABLE]
Following Gasper and Rahman [3], define the bilateral basic hypergeometric series by
[TABLE]
When , the bilateral basic hypergeometric series becomes the unilateral basic hypergeometric series
[TABLE]
Then Ramanujan’s summation formula (cf. [3, Appendix II. 29]) and Bailey’s summation formula (cf. [3, Appendix II. 33]) can be stated as
[TABLE]
where ,
[TABLE]
provided .
Via the analytic continuation method, Ismail [4] gave a simple proof of (5). Askey and Ismail [2] confirmed (8) in the same way. More results from this method can be seen in Zhu [7].
Inspired by the work just mentioned, we shall establish the following three transformation formulas involving bilateral basic hypergeometric series through the analytic continuation method.
Theorem 1**.**
Let be complex numbers such that . Then
[TABLE]
Theorem 2**.**
Let be complex numbers such that . Then
[TABLE]
Theorem 3**.**
Let be complex numbers with and . Then
[TABLE]
[TABLE]
Fixing in Theorem 1, we obtain the following result.
Corollary 4**.**
Let be complex numbers such that . Then
[TABLE]
Setting in Theorem 2, we get the following conclusion.
Corollary 5**.**
Let be complex numbers such that . Then
[TABLE]
When , Theorem 3 becomes
[TABLE]
Calculating the series on the right-hand side by Rogers’ summation formula (cf. [3, Appendix II. 21]):
[TABLE]
we arrive at Bailey’s summation formula (8).
By means of Cauchy’s method, Jouhet [5] found the identity involving two series and two series:
[TABLE]
where , and .
Recall the transformation formula (cf. [3, Appendix III. 23]):
[TABLE]
with and the three-term relation (cf. [3, Exercise 2.16]):
[TABLE]
where
[TABLE]
On the basis of the last two equations, it is not difficult to understand the equivalence of Theorem 3 and Fouhet’s result (26).
The rest of the paper is arranged as follows. The proofs of Theorems 1 and 2 will respectively be displayed in Sections 2 and 3. In section 4, we shall give a new proof of Theorem 3.
2. Proof of Theorem 1
In order to prove Theorem 1, we need the following two lemmas.
Lemma 6**.**
For all integers , it holds
[TABLE]
Lemma 7** **(cf. [6, p. 90]; see also [7, p.
741]).
Let be a connected open set and , be analytic on . If and agree infinitely often near an interior point of , then we have for all .
Proof of Theorem 1.
A known transformation formula for unilateral basic hypergeometric series (cf. [3, Equation (3.4.7)]) reads
[TABLE]
Performing the replacements in this equation, we obtain
[TABLE]
Let be a nonnegative integer throughout the paper. On one hand, it is easy to deduce, from Lemma 6, the following relation:
[TABLE]
On the other hand, we derive, from Lemma 6, the following relation:
[TABLE]
[TABLE]
Substituting (54) and (65) into (46) leads us to
[TABLE]
Define two functions and to be
[TABLE]
[TABLE]
Then (68) shows that
[TABLE]
for . Based on Lemma 7, (74) holds for all . By the analytic continuation method, the restriction on can by relaxed. Thus we get
[TABLE]
Replying by in it, we arrive at Theorem 1. ∎
3. Proof of Theorem 2
Now we begin to prove Theorem 2 with the help of Lemma 7 and the following transformation formula for unilateral basic hypergeometric series (cf. [3, Equation (3.4.8)]):
[TABLE]
Proof of Theorem 2.
Employing the replacements in the last equation, we obtain
[TABLE]
According to Lemma 6, we can reformulate it as
[TABLE]
This transformation tells us that Theorem 2 is true for . Therefore, we can prove Theorem 2 via Lemma 7 and the analytic continuation argument. ∎
4. A new Proof of Theorem 3
For the sake of proving Theorem 3, we require the following lemma.
Lemma 8**.**
Let be complex numbers. Then
[TABLE]
where and .
Proof.
It is clear from (38) that
[TABLE]
Putting it into the transformation formula (cf. [3, Appendix III. 37]):
[TABLE]
and then interchanging the parameters and , we decuce Lemma 8. ∎
Proof of Theorem 3.
Performing the replacements , in Lemma 8, we derive
[TABLE]
By means of Lemma 6, the last transformation can be manipulated as
[TABLE]
This indicates that Theorem 3 is true for . Thus we have proved Theorem 3 through Lemma 7 and the analytic continuation argument. ∎
Acknowledgments
The work is supported by the National Natural Science Foundations of China (Nos. 12071103 and 11661032).
The reference list from the paper itself. Each links out to its DOI / PubMed record.
- 1[1]
- 2[2] R. Askey, M.E.H. Ismail, The very well poised ψ 6 6 subscript subscript 𝜓 6 6 {}_{6}\psi_{6} , Proc. Amer. Math. Soc. 77 (1979), 218–222.
- 3[3] G. Gasper, M. Rahman, Basic Hypergeometric Series ( ( ( 2nd edition ) ) ) , Cambridge University Press, Cambridge, 2004.
- 4[4] M.E.H. Ismail, A simple proof of Ramanujan’s ψ 1 1 subscript subscript 𝜓 1 1 {}_{1}\psi_{1} sum, Proc. Amer. Math. Soc. 63 (1977), 185–186.
- 5[5] F. Jouhet, Some more Semi-finite forms of bilateral basic hypergeometric series, Ann. Combin. 11 (2007), 47–57.
- 6[6] S. Lang, Complex Analysis ( ( ( 4nd edition ) ) ) , Springer, New York, 1999.
- 7[7] J. Zhu, Generalizations of a terminating summation formula of basic hypergeometric series and their applications, J. Math. Anal. Appl. 436 (2016), 740–747.