/ / Article
Special Issue

## Recent Theory and Applications on Numerical Algorithms and Special Functions

View this Special Issue

Research Article | Open Access

Volume 2014 |Article ID 714560 | 5 pages | https://doi.org/10.1155/2014/714560

# Certain Class of Generating Functions for the Incomplete Hypergeometric Functions

Accepted17 Jul 2014
Published18 Aug 2014

#### Abstract

Generating functions play an important role in the investigation of various useful properties of the sequences which they generate. In this paper, we aim to establish certain generating functions for the incomplete hypergeometric functions introduced by Srivastava et al. (2012). All the derived results in this paper are general and can yield a number of (known and new) results in the theory of generating functions.

#### 1. Introduction and Definitions

A lot of research work has recently come up on the study and development of the familiar incomplete Gamma type functions like and given in and , respectively. The study of incomplete Gamma functions has a very long history (see, e.g., ) and now stands on fairly firm footing through the research contributions of various authors (see, e.g., ). Incomplete Gamma functions are important special functions and their closely related ones are widely used in physics and engineering; therefore, they are of interest to physicists, engineers, statisticians, and mathematicians.

The theory of the incomplete Gamma functions, as a part of the theory of confluent hypergeometric functions, has received its first systematic exposition by Tricomi  in the early 1950s. The familiar incomplete Gamma functions and are defined, respectively, by

The following decomposition formula holds: where is the familiar Gamma function defined by

Historically, and were first studied in 1877 for by Prym . The functions and are also referred to as Prym’s functions. For general (even for ), the function appears in Exercises de Calcul Integral by Legendre  and in some of his later works.

The function can be expressed in terms of Tricomi’s confluent hypergeometric function as follows (see [6, page 266, Equation ]):

In terms of the Gamma function , Pochhammer symbol is defined (for ) by (see, e.g., [10, page 2 and pages 4–6]) where and denote the sets of complex numbers and nonpositive integers, respectively.

Recently, Srivastava et al.  introduced and studied some fundamental properties and characteristics of a family of the following two potentially useful generalized incomplete hypergeometric functions defined as follows: where and are certain interesting generalizations of the Pochhammer symbol which are defined, in terms of the incomplete Gamma type functions and given in and , by

These incomplete Pochhammer symbols and , which were defined by Srivastava et al. , like , also satisfy the following decomposition relation:

Remark 1. As already mentioned by Srivastava et al. [16, Remark 7] (see also [17, page 3220, Remark]), since the precise (sufficient) conditions under which the infinite series in the definitions and would converge absolutely can be derived from those that are well-documented in the case of the generalized hypergeometric function () (see, for details, [20, pages 72-73] and [11, page 20]; see also ). Indeed, in their special case when , both () and () would reduce immediately to the extensively investigated generalized hypergeometric function () (see, e.g., [20, Chapter 5]; see also [10, Section 1.5]). Furthermore, as an immediate consequence of the definitions and , we have the following decomposition formula: in terms of the familiar generalized hypergeometric function ().

Generating functions play an important role in the investigation of various useful properties of the sequences which they generate. They are used in finding certain properties and formulas for numbers and polynomials in a wide variety of research subjects, indeed, in modern combinatorics. For a systematic introduction to, and several interesting (and useful) applications of, the various methods of obtaining linear, bilinear, bilateral, or mixed multilateral generating functions for a fairly wide variety of sequences of special functions (and polynomials) in one, two, and more variables, among much abundant literature, we refer to the extensive works by Srivastava and Manocha  and Agarwal and Koul . In this regard, in fact, a remarkably large number of generating functions involving a variety of special functions have been developed by many authors (see, e.g., [24, 26]; see also ). Also many generating functions containing the incomplete hypergeometric functions and have been presented (see, e.g., [17, Corollary 3]). Here, motivated mainly by the works of both Chen and Srivastava  and Srivastava and Cho , we present certain generating functions involving the incomplete hypergeometric functions and . Furthermore, it should be mentioned in passing that our results in the present paper are established by using a different method employed by .

#### 2. Generating Functions for the Incomplete Hypergeometric Functions

In this section, we establish certain generating functions for the incomplete hypergeometric functions and asserted by Theorem 2.

Theorem 2. The following generating functions hold true:

Proof. For convenience, let the left-hand side of be denoted by . Applying the series expression of to , we get Using the following known identities (see, e.g., [10, page 5]): being the set of integers and , we can prove the following identity (see [28, page 169]): By changing the order of summations in and using the identity , after little simplification, we have We find that the inner sum in is the generalized binomial expansion Finally, replacing the inner sum of by the identity yields our desired result .
It is easy to see that a similar argument as in the proof of will establish the result . This completes the proof of Theorem 2.

Remark 3. Recently, Srivastava and Cho  presented a very general class of certain interesting generating functions involving the incomplete hypergeometric functions and by essentially using the following interesting and useful unified expansion formula given by Gould (see [29, page 196, Equation ]; see also [17, page 3221]): where , , and are complex numbers independent of and is a function of defined implicitly by

The results [17, Corollary 3] look very similar to those given in Theorem 2. Yet, it is easy to see that they cannot be special or general cases of the other one’s results.

#### 3. Further Generalization of the Generating Functions for the Incomplete Hypergeometric Functions

A further generalization of the incomplete hypergeometric functions and is given in the following definition.

Definition 4. Let us introduce two sequences and defined by where, for convenience, abbreviates the array of parameters as follows:
Then, as in Theorem 2, we can give the following generating functions for the generalized incomplete hypergeometric functions asserted by Theorem 5.

Theorem 5. Each of the following identities holds true:

Proof. Similarly as in Theorem 2, we can prove the results in Theorem 5. So their details are left to the interested reader by, instead of the essential identity , presenting the following identity:

It should be noted that, if we set and replace by in , we are easily led to the result .

Concluding Remarks. If we add the two generating functions and and use the decomposition formula , we have an interesting result expressed in terms of generalized hypergeometric functions :

We also observe that the result corresponds to that given in [28, page 170, Equation ].

The generalized incomplete hypergeometric functions given in and reduce, when , to the generalized hypergeometric function () whose particular cases are known to express most of the special functions occurring in the mathematical, physical, and engineering sciences. Therefore most of the known and widely investigated special functions are expressible also in terms of the generalized incomplete hypergeometric functions () and () (for some interesting examples and applications, see [16, Sections 5 and 6]). In view of this observation, the results presented here, being of general character, can yield numerous generating functions for a certain class of incomplete hypergeometric polynomials (see ) and other special functions which are expressible in terms of hypergeometric functions. Finally, we conclude our present investigation by remarking that our results presented here are also believed to give some contribution to the communication theory, probability theory, and groundwater pumping modeling.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

The authors should express their deep gratitude to all the referees for their very helpful and critical comments originating from only detailed reviews of this paper by sharing their valuable time. This research was, in part, supported by the Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology of the Republic of Korea (Grant no. 2010-0011005). This work was supported by Dongguk University Research Fund.

1. F. E. Prym, “Zur theorie der gamma function,” The Journal für die Reine und Angewandte Mathematik, vol. 82, pp. 165–172, 1877. View at: Google Scholar
2. M. Abramowitz and I. A. Stegun, Eds., Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables, Applied Mathematics Series 55, National Bureau of Standards, Washington DC, USA, 9th edition, 1984. View at: MathSciNet
3. P. Agarwal, “Certain properties of the generalized Gauss hypergeometric functions,” Applied Mathematics & Information Sciences, vol. 8, no. 5, pp. 2315–2320, 2014. View at: Google Scholar
4. G. E. Andrews, R. Askey, and R. Roy, Special Functions, vol. 71 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, UK, 1999. View at: Publisher Site | MathSciNet
5. E. H. Doha and W. M. Abh-Elhameed, “New linearization formulae for the products of Chebyshev polynomials of third and fourth kind,” Rocky Mountain Journal of Mathematics. In press. View at: Google Scholar
6. A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions, vol. 1, McGraw-Hill Book Company, New York, NY, USA, 1953. View at: MathSciNet
7. N. L. Johnson, S. Kotz, and N. Balakrishnan, Continuous Univariate Distributions, vol. 2, John Wiley & Sons, New York, NY, USA, 1995. View at: MathSciNet
8. W. Koef, Hypergepometric Summation Vieweg, Braunschweig-Wiebaden, 1998.
9. H. M. Srivastava and J. Choi, Series Associated with the Zeta and Related Functions, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2001. View at: Publisher Site | MathSciNet
10. H. M. Srivastava and J. Choi, Zeta and q-Zeta Functions and Associated Series and Integrals, Elsevier Science, Amsterdam, The Netherlands, 2012.
11. H. M. Srivastava and P. W. Karlsson, Multiple Gaussian Hypergeometric Series, Ellis Horwood, Chichester, UK, 1985.
12. H. M. Srivastava and B. R. K. Kashyap, Special Functions in Queuing Theory and Related Stochastic Processes, Academic Press, New York, NY, USA, 1982. View at: MathSciNet
13. N. M. Temme, Special Functions: An Introduction to Classical Functions of Mathematical Physics, John Wiley and Sons, New York, NY, USA; Brisbane and Toronto, Chichester, UK, 1996. View at: Publisher Site | MathSciNet
14. G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, UK, 2nd edition, 1944. View at: MathSciNet
15. E. T. Whittaker and G. N. Watson, A Course of Modern Analysis: An Introduction to the General Theory of Infinite Processes and of Analytic Functions; With an Account of the Principal Transcendental Functions, Cambridge University Press, New York, NY, USA, 4th edition, 1963.
16. H. M. Srivastava, M. A. Chaudhry, and R. P. Agarwal, “The incomplete Pochhammer symbols and their applications to hypergeometric and related functions,” Integral Transforms and Special Functions, vol. 23, no. 9, pp. 659–683, 2012. View at: Publisher Site | Google Scholar | MathSciNet
17. R. Srivastava and N. E. Cho, “Generating functions for a certain class of incomplete hypergeometric polynomials,” Applied Mathematics and Computation, vol. 219, no. 6, pp. 3219–3225, 2012. View at: Publisher Site | Google Scholar | MathSciNet
18. F. G. Tricomi, “Sulla funzione gamma incompleta,” Annali di Matematica Pura ed Applicata, vol. 31, no. 1, pp. 263–279, 1950. View at: Publisher Site | Google Scholar | MathSciNet
19. A. M. Legendre, Exercises de Calcul Intégral Sur Divers Orders de Transcendantes et Sur les Quadratures, vol. 1, Courcier, Paris, France, 1981.
20. E. D. Rainville, Special Functions, Macmillan, New York, NY, USA, 1960.
21. B. C. Carlson, Special Functions of Applied Mathematics, Academic Press, New York, NY, USA, 1977. View at: MathSciNet
22. Y. L. Luke, Mathematical Functions and Their Approximations, Academic Press, New York, NY, USA, 1975. View at: MathSciNet
23. L. J. Slater, Generalized Hypergeometric Functions, Cambridge University Press, Cambridge, UK, 1966. View at: MathSciNet
24. H. M. Srivastava and H. L. Manocha, A Treatise on Generating Functions, Halsted Press, Ellis Horwood, John Wiley & Sons, New York, NY, USA, 1984.
25. P. Agarwal and C. L. Koul, “On generating functions,” Journal of Rajasthan Academy of Physical Sciences, vol. 2, no. 3, pp. 173–180, 2003.
26. D. Zeitlin, “A new class of generating functions for hypergeometric polynomials,” Proceedings of the American Mathematical Society, vol. 25, pp. 405–412, 1970. View at: Google Scholar | MathSciNet
27. H. M. Srivastava, “Some bilateral generating functions for a class of special functions I and II,” Indagationes Mathematicae (Proceedings), vol. 83, no. 2, pp. 234–246, 1980. View at: Google Scholar
28. M.-P. Chen and H. M. Srivastava, “Orthogonality relations and generating functions for Jacobi polynomials and related hypergeometric functions,” Applied Mathematics and Computation, vol. 68, no. 2-3, pp. 153–188, 1995. View at: Publisher Site | Google Scholar | MathSciNet
29. H. W. Gould, “A series transformation for finding convolution identities,” Duke Mathematical Journal, vol. 28, pp. 193–202, 1961. View at: Publisher Site | Google Scholar | MathSciNet

We are committed to sharing findings related to COVID-19 as quickly and safely as possible. Any author submitting a COVID-19 paper should notify us at help@hindawi.com to ensure their research is fast-tracked and made available on a preprint server as soon as possible. We will be providing unlimited waivers of publication charges for accepted articles related to COVID-19. 