Discrete Analogues of the Erdélyi Type Integrals for Hypergeometric Functions

Vyas, Yashoverdhan; Bhatnagar, Anand V.; Fatawat, Kalpana; Suthar, D. L.; Purohit, S. D.

doi:https://doi.org/10.1155/2022/1568632

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Research Article | Open Access

Volume 2022 | Article ID 1568632 | https://doi.org/10.1155/2022/1568632

Discrete Analogues of the Erdélyi Type Integrals for Hypergeometric Functions

Yashoverdhan Vyas,¹Anand V. Bhatnagar,¹Kalpana Fatawat,²D. L. Suthar,³and S. D. Purohit⁴

Academic Editor: Serkan Araci

Received05 Jul 2022

Accepted22 Aug 2022

Published07 Oct 2022

Abstract

Gasper followed the fractional calculus proof of an Erdélyi integral to derive its discrete analogue in the form of a hypergeometric expansion. To give an alternative proof, we derive it by following a procedure analogous to a triple series manipulation-based proof of the Erdélyi integral, due to “Joshi and Vyas”. Motivated from this alternative way of proof, we establish the discrete analogues corresponding to many of the Erdélyi type integrals due to “Joshi and Vyas” and “Luo and Raina” in the form of new hypergeometric expansion formulas. Moreover, the applications of investigated discrete analogues in deriving some expansion formulas involving orthogonal polynomials of the Askey-scheme and a new generalization of Whipple’s transformation for a balanced ₄F₃ in the form of an transformation, are also discussed.

1. Introduction, Motivation, and Preliminaries

The Gauss hypergeometric function ([1], p. 243, Equation (III.3)), see also ([2], p. 18, Equation (17)):is an instance of the following generalized hypergeometric function ([2], p. 19, Equation (23)):where can be either 0 or any natural number. The parameter vector isin which all the parameters can take any real or complex value under the restriction that can neither be zero nor a negative integer. The series in (2) converges or diverges in accordance with the value of less than or greater than 1, respectively. Further, if we takethen for , the series in (2) converges or diverges in accordance with the value of is less than or greater than 0, respectively. For a detailed discussion on convergence of (2), we refer to ([2], p. 20) and ([3], p. 43).The symbol in (2) is the well-known symbol of Pochhammer defined byAlso, we haveFor further such relations, we refer to ([1], pp. 239–240, Appendix I).The following resultwhere , is known as Euler’s integral ([4], p. 47, Theorem 16). An extension of (8) was developed by Bateman [5] which is as follows:where .About eighty years ago, Erdélyi [6] gave the extensions of Euler’s integral (8) and Bateman’s integral (9), in the form of following equations (10)–(12), as given in the following equation:where ,where , andwhere . Further, equations (8)–(12) satisfy the constraints and . The derivations of above equations in [6] were based on the fractional calculus. For some important applications of (10), as shown in [7].

Thirty five years hence Erdélyi [6], Gasper [7] derived the discrete analogue of Erdélyi’s integral (10) as given in the following equation ([7], Equation (27)):Gasper [7] proved (13) by following the steps analogous to Erdélyi’s fractional calculus proof of (10). He used three transformation formulas ([7], Equations (22)–(24)) to prove (13), where one of the transformation formulas ([7], Equations (23)) is as follows:As explained in Gasper ([7], p. 208) and ([8], p. 2), replacing by , , respectively, in (13) and choosing through integer values of with fixed such that lies between 0 and 1, and finally using analytical continuation with respect to , we get (10). For this conversion, we utilize the fundamental definition of definite integral and a limit relation involving Pochhammer symbols ([4], p. 11). Moving a few steps ahead, Gasper [8] studied expansions to provide generalizations of (13) and other results, along with the applicability to orthogonal polynomials [7, p. 6]. Lievens [9] discussed the association of such expansions with the invariant group of the Lie algebra. The expansions given by [8] were specialized to give expansion formulas for the Racah polynomials, the Askey–Wilson polynomials and their -analogues, where the Racah polynomials ([10], p. 25, Equation (1.2.1)) and the Askey–Wilson polynomials ([10], p. 23, Equation (1.1.1)) are defined as follows:with a positive integer.As a matter of fact, the further generalizations of these expansions were developed by Joshi and Vyas [11], in the form of and expansions, along with the applicability to biorthogonal rational functions ([11], p. 221, Section 3).

Many researchers, for example, [7, 8, 11–19] have studied and investigated several kinds of integrals that involve and represent the hypergeometric functions, on account of the numerous applications of these integrals, for example, as shown in [13].

Following, Erdélyi [6] and Gasper [7, 8]; Joshi and Vyas [15] gave the alternative proofs of three integrals of Erdélyi by using the series rearrangement method ([4], see also [3]) and the three classical summation theorems, namely, the Gauss, Vandermonde, and Saalschütz summation theorems. Motivated from the above way of proof, they investigated many new integrals for certain , which were similar to the Erdélyi integrals due to the inclusion of the powers and hypergeometric functions and hence named, Erdélyi type integrals. Furthermore, they obtained many other varieties of Erdélyi’s integrals, for example ([15], p. 132, Equations (4.1), (4.4), (4.5), (4.7) to (4.9) and (5.2)). In the sequel, they [16] also discussed -extensions of the Erdélyi type integrals, by using series rearrangement. Certain classical series rearrangements were also used by [20, 21] to derive extensions of the Bailey transform, which in turn, were applied to establish remarkable ordinary and -hypergeometric identities.

Recently, Luo and Raina [22] have obtained two new Erdélyi type integrals ([22], Theorems 3.1 and 3.2), which are extensions of (10). They further investigated the generalization of Erdélyi type integral ([15], p. 132, Equation (4.1)) using the extended version of Saalschütz summation theorem [23]:where are the non-zero roots of the associated parametric equation of degree , wherewithThe hypergeometric series is a polynomial in of degree where . The coefficients are defined bywhere stands for the second kind Stirling number ([24], Chapter 6). The coefficients can be obtained from the following generating relation:

The application of one of the extended Erdélyi type integrals due to ([22], Theorem 3.2) in deriving a generalization of Thomae-type transformation for with integral parameter differences ([22], p. 11, Equation (4.6)) is also discussed.

However, Gasper [7, 8] developed the discrete analogues for Erdélyi’s integrals but, the discrete analogues corresponding to the Erdélyi type integrals due to ([22], p. 9, Section 4) and ([15], Equations (4.1) and (3.1) to (3.7)) have not been investigated till now.

It will be our endeavor in this paper to conjecture and prove the discrete analogues of the Erdélyi type integrals due to [15, 22] in the form of new hypergeometric expansion formulas by following the method analogous to [15], that is, the utilization of series manipulation techniques in conjunction with certain classical summation theorems, namely, the Vandermonde theorem ([1], p. 243, Equation (III.4)), the Saalschütz summation theorem ([1], p. 243, Equation (III.2)), the extended Saalschütz summation theorem [23], the Dougall theorems for ([1], p. 244, Equation (III.13)) and the ([1], p. 244, Equation (III.14)), and the transformation formula (14). To avoid repetition, the citation details for the well-known summation theorems mentioned above will be omitted in the remaining part of the paper.

Our primary aim here is to develop the discrete analogues of the Erdélyi type integrals due to Joshi and Vyas [15] and Luo and Raina [22]. The organization of the current research work is as follows.

In Section 2, a step-by-step demonstration of the alternative proof of the hypergeometric expansion or discrete analogue (13) of the Erdélyi integral, along the lines of a triple series manipulation-based derivation of the Erdélyi’s integral (10) by Joshi and Vyas [15], is detailed. In Section 3, the statements and derivations of the discrete analogues of the Erdélyi type integrals due to [15, 22] in the form of new hypergeometric expansion theorems, are presented. In Section 4, some remarkable applications of the investigated results are discussed, where the application of Theorem 2 provides the expansion formulas involving the orthogonal polynomials (Racah and Askey–Wilson). Further, the application of Theorem 1 enable us to develop a generalization of the well-known Whipple’s transformation of into ([25], p. 140, Theorem 3.3.3), in the form of a transformation of into . This transformation of into leads to an extended Thomae-type transformation due to Luo and Raina ([22], Equation (4.6)) (see also [26], p. 114, Theorem 2) as a special case. Thus, this paper shows the efficiency of the series rearrangement technique in deriving the discrete analogues of the integrals due to ([22], p. 9) and ([15], Equations (4.1) and (3.1) to (3.7)) with the applications.

2. Alternative Proof for the Discrete Analogue of Erdélyi’s Integral

In the present section, we are providing the alternative proof of (13) by using the series manipulation techniques, the transformation formula (14), the Vandermonde sum, and the Saalschütz sum.

Proof. Step 1: Denoting the right-side of (13) by and writing the ’s in their series form, we get Step 2: Next, by applying the following double series manipulation lemma [3, 4], we get on equation (23) and then taking an inner series in , we can write Step 3: Now, replacing and in (14) and applying the resulting transformation on of the right side of (27), we can obtain Step 4: Next, on applying the following triple series rearrangement: to the triple series (27), and then taking an inner series in , we obtain Step 5: Now, applying Vandermonde sum to simplify the present in (29) and then taking an inner series in , we can write Finally, the application of the Saalschütz sum and further simplifications lead to the left side of (13).

Remark 1. In the next Section 3, we shall prove Theorems 1 and 5 to 10 by following the steps similar to those we have elaborated in the present Section. It may be noted that Theorems 2–4 follow as special cases of Theorems 1 and 3.

3. Discrete Analogues of the Erdélyi Type Integrals

Here, we state and prove a number of new hypergeometric expansions as discrete analogues for the Erdélyi type integrals due to ([22], pp. 9–10, Equation (4.1)) and ([15], Equations (4.1) and (3.1) to (3.7)) in the form of Theorems 1, 2, and 4 to 10, respectively. We also obtain an additional expansion as Theorem 3, which is the generalization of Theorem 4.

Theorem 1. The following assertion holds true:where the lower parameters have a similar interpretation as given by equations (18)–(22) provided the following replacements are applied:

Proof. The proof of the above expansion follows the same steps as discussed in Section 2, except using the Vandermonde and the extended Saalschütz sum (Equations (17)–(22)), one by one, to simplify the appearing inner series.

Remark 2. To convert Theorem 1 into the Erdélyi type integral ([22], pp. 9–10, Equation (4.1)), first, we replace by , respectively and then, we choose through integer values of with fixed such that lies between 0 and 1. Finally, we use equations (5)–(7), a limit relation involving Pochhammer symbols ([4], p. 11), and analytical continuation with respect to to get the desired result. The same procedure may be applied to convert Theorems 2, 4–10) to their corresponding Erdélyi type integrals ([15], Equations (4.1), and (3.1) to (3.7)), respectively.

Theorem 2. The following assertion holds true:

Proof. It can be easily observed that the assertion in Theorem 2 follows from Theorem 1 when .

Theorem 3. The following assertion holds true:

Proof. The following replacements arein Theorem 1 lead us to Theorem 3.

Theorem 4. The following assertion holds true:

Proof. The case of Theorem 3 leads us to the expansion formula given in Theorem 4.

Theorem 5. The following assertion holds true:

Proof. The proof of above theorem follows the steps explained in Section 2 and utilizes the Saalschütz sum twice.

Theorem 6. The following assertion holds true:

Proof. Following the steps discussed in Section 2 and applying the Vandermonde and the Dougall sum, one by one, we can easily get the assertion stated in Theorem 6.

Theorem 7. The following assertion holds true:

Proof. To prove the above assertion, we follow the method discussed in Section 2, and apply the Vandermonde and the Saalschütz sum.

Theorem 8. The following assertion holds true:

Proof. Following the steps mentioned in Section 2 and using the Vandermonde and the Dougall sum, we get the desired expansion.

Theorem 9. The following assertion holds true:

Proof. To prove the above assertion we follow the similar steps as given in Section 2 and will use the Saalschütz and the Dougall sum.

Theorem 10. The following assertion holds true:

Proof. The proof of Theorem 10 follows the steps explained in Section 2 and will make use of the Vandermonde and the Dougall sum.

4. Applications

In this Section, we shall apply Theorem 2 to investigate the expansion formulas involving the Racah and the Askey–Wilson polynomials and product of hypergeometric functions. Further, Theorem 1 will be utilized to derive the generalized Whipple transformation of into by using the Saalschütz sum and then, applying the double series manipulation (24) and the Vandermonde sum.(1)On applying the following replacements and then, in Theorem 2 lead to an expansion for the Racah polynomials as where and one of the denominator parameters out of the three that is, , , and in should be equal to .(2)The following replacements and then , , in Theorem 1 lead to an expansion for the Racah polynomials as subject to the restrictions similar to those mentioned with equation (44).(3)Next, letting and in Theorem 2, we get the following expansion for the Askey-Wilson polynomials (4)In Theorem 1, we can apply the Saalschütz sum to simplify the series under the following condition: which converts Theorem 1 into the following result: Next, expressing the in its series form, applying the double series manipulation lemma (24), taking an inner series in and then, summing this series by the Vandermonde theorem, we get The equation (52) is a generalization of the Whipple transformation of into mentioned in ([25], p. 140, Theorem 3.3.3), which follows from (52) when .(5)Now, replacing by in (52) and then taking a limit as , it leads us to an extended Thomae-type transformation due to Luo and Raina ([22], Equation (4.6)) see also Kim et al. ([23], p. 114, Theorem 2). At this point, it may be observed that ([22], Equation (4.6)) is equivalent to ([23], p. 114, Theorem 2).

5. Conclusion

This research paper clearly shows that how the series manipulation technique analogous to Joshi and Vyas [15] can effectively be applied to derive many new hypergeometric expansion formulas as discrete analogues for the Erdélyi type integrals. Further, the derived discrete analogues of this paper are also applied to provide the expansion formulas involving the Racah and the Askey–Wilson polynomials and a very important generalization of the Whipple transformation of into in the form of a transformation of into . Moreover, it will not be out of place to observe here that Gasper [8] investigated the -analogues and generalizations of the discrete analogues of the three integrals of Erdélyi to provide new expansion formulas for and along with the applicability to orthogonal polynomials and some new double Kampé de Fériet summations, which in turn were further extended by Joshi and Vyas [11] in the form of and expansions, along with the applicability to biorthogonal rational functions. Some future research directions on this work can be the investigation of further generalizations of the expansion formulas of this paper along the lines of [8, 11]. A work along these directions is going on and in the foreseeable future, we may hope to submit these new results for publication.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this article.

Authors’ Contributions

The authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.

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Copyright © 2022 Yashoverdhan Vyas 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.

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