Special Issue

## Recent Trends in Special Functions and Analysis of Differential Equations

View this Special Issue

Research Article | Open Access

Volume 2020 |Article ID 8596736 | https://doi.org/10.1155/2020/8596736

Shilpi Jain, Juan J. Nieto, Gurmej Singh, Junesang Choi, "Certain Generating Relations Involving the Generalized Multi-Index Bessel–Maitland Function", Mathematical Problems in Engineering, vol. 2020, Article ID 8596736, 5 pages, 2020. https://doi.org/10.1155/2020/8596736

# Certain Generating Relations Involving the Generalized Multi-Index Bessel–Maitland Function

Accepted04 Aug 2020
Published25 Aug 2020

#### Abstract

Generating relations involving the special functions have already proved their important role in mathematics and other fields of sciences. In this paper, we aim to provide some presumably new generating relations in connection with the generalized multi-index Bessel–Maitland function . The main results presented here, being very general, can yield a number of particular or equivalent identities, some of which are explicitly demonstrated.

#### 1. Introduction and Preliminaries

Here and elsewhere, let , , , , and be the sets of complex numbers, real numbers, positive real numbers, positive integers, and nonpositive integers, respectively.

The Bessel–Maitland function is defined as (see Marichev [1])

Pathak [2] gave the following more generalized form of generalized Bessel–Maitland function (1):

Remark 1. Even though Pathak excluded in (2), the case yields (1).
If , , is replaced by , and is replaced by in (2), then generalized Bessel–Maitland function reduces to the Mittag–Leffler function which was studied by Wiman [3] as follows:If is replaced by and is replaced by in (2), then the generalized Bessel–Maitland function reduces to the well-known generalized Mittag–Leffler function which was introduced by Shukla and Prajapati [4] as follows:Jain and Agarwal [5] generalized Bessel–Maitland function (1) as follows:Choi and Agarwal [6] investigated the following generalized multi-index Bessel function:where and , , , , and such that

Remark 2. It is easily found that generalized multi-index Bessel–Maitland function (9) is equivalent to the generalized multi-index Mittag–Leffler function defined and studied by Saxena and Nishimoto [7] (see also [8]).
Pohlen [9] introduced the Hadamard product (or the convolution) of two analytic functions and as follows:where . Here, and are analytic at whose Maclaurin series with their respective radii of convergence and areThe concept of the Hadamard product has turned out to be useful, particularly, in factorizing a newborn function, which is usually expressed as a Maclaurin series, into two known functions (see, e.g., [1013]).
The -th derivative of the function is easily found to be given in terms of gamma function as follows:Generating functions have been widely used in exploring certain properties and formulas involving sequences and polynomials in a wide range of research subjects. Many researchers have developed a remarkably large number of generating functions associated with a variety of special functions. For some works on this subject, one may refer, for example, to an extensive monograph [1425] and the literature cited therein. In this search, we aim to provide some presumably new generating relations in connection with generalized multi-index Bessel–Maitland function (9). The main results developed here, being very general, can be reduced to produce a large number of presumably new and potentially useful generating relations for other known functions, some of which are demonstrated.

#### 2. Generating Relations

We give two generating relations involving generalized multi-index Bessel–Maitland function (9) asserted by the following theorems.

Theorem 1. Let and , , , , and such thatAlso, let . Then,

Proof. We replace by in the left-hand side of (15) and denote the resulting expression by . Then, using form (9), on expanding the function in series, givesDifferentiating times both sides of (16) with respect to with the aid of (13) (term-by-term differentiation can be verified under the given conditions), we findwhich is simplified to yieldDecomposing series (18) into Hadamard product (11), we obtainExpanding as the Taylor series givesCombining (16), (19), and (20), we obtainFinally, setting yields desired result (15).

Theorem 2. Let and , , , , and such thatAlso, let . Then,

Proof. Let be the left-hand side of (23). Using (9), on expanding the function in series, givesInterchanging the order of summations in (24) and using the known identity (see, e.g., [26, p. 5])we haveUsing the generalized binomial expansion, we find that the inner sum in (26) givesFinally, interpreting (26) with the help of (27) yields desired result (23).

#### 3. Further Remarks

Here, we choose to give some equivalent identities and particular cases of the results in Theorems 1 and 2. As noted in Remark 2, setting by and by in (15) and (23) gives two corresponding generating relations involving the generalized multi-index Mittag–Leffler function , which are asserted, respectively, in Corollaries 1 and 2.

Corollary 1. Let and , , , , and such thatAlso, let . Then,

Corollary 2. Let and , , , , and such thatAlso, let . Then,

The particular cases of (15), (23), (29), and (31) when give the following generating relations, stated, respectively, in Corollaries 36.

Corollary 3. Let , , , , and such that , , , and . Also, let . Then,

Corollary 4. Let , , , , and such that , , , and . Also, let . Then,

Corollary 5. Let , , , , and such that , , , and . Also, let . Then,

Corollary 6. Let , , , , and such that , , , and . Also, let . Then,

#### 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 paper.

#### Acknowledgments

This research has been partially supported by the Agencia Estatal de Investigacion (AEI) of Spain, co-financed by the European Fund for Regional Development (FEDER) corresponding to the 2014–2020 multiyear financial framework (project MTM2016-75140-P), and also supported by Xunta de Galicia (grant ED431C 2019/02). Shilpi Jain also thanks SERB (project number: MTR/2017/000194) for providing necessary facility.

#### References

1. O. I. Marichev, Handbook of Integral Transforms and Higher Transcendental Functions, Ellis Horwood, Chichester, UK, 1983.
2. R. S. Pathak, “Certain convergence theorems and asymptotic properties of a generalization of Lommel and Bessel-Maitland transform,” Proceedings of the National Academy of Sciences, India, Section A, vol. 36, pp. 81–86, 1996. View at: Google Scholar
3. A. Wiman, “Über de fundamental satz in der theorie der funktionen .,” Acta Mathematica, vol. 29, pp. 191–201, 1905. View at: Publisher Site | Google Scholar
4. A. K. Shukla and J. C. Prajapati, “On a generalization of Mittag-Leffler function and its properties,” Journal of Mathematical Analysis and Applications, vol. 336, no. 2, pp. 797–811, 2007. View at: Publisher Site | Google Scholar
5. S. Jain and P. Agarwal, “A new class of integral relations involving general class of polynomials and -functions,” Walailak Journal of Science and Technology, vol. 12, no. 11, pp. 1009–1018, 2015. View at: Google Scholar
6. J. Choi and P. Agarwal, “A note on fractional integral operator associated with multiindex Mittag-Leffler functions,” Filomat, vol. 30, no. 7, pp. 1931–1939, 2016. View at: Publisher Site | Google Scholar
7. R. K. Saxena and K. Nishimoto, “-fractional calculus of generalized Mittag-Leffler functions,” Journal of Fractional Calculus, vol. 37, pp. 43–52, 2010. View at: Google Scholar
8. R. K. Saxena, T. K. Pogány, J. Ram, and J. Daiya, “Dirichlet averages of generalized multi-index Mittag-Leffler functions,” Armenian Journal of Mathematics, vol. 3, no. 4, pp. 174–187, 2010. View at: Google Scholar
9. T. Pohlen, The Hadamard Product and Universal Power Series, Universität Trier, Trier, Germany, 2009.
10. P. Agarwal and J. Choi, “Fractional calculus operators and their image formulas,” Journal of the Korean Mathematical Society, vol. 53, no. 5, pp. 1183–1210, 2016. View at: Publisher Site | Google Scholar
11. V. Kiryakova, “On two Saigos fractional integral operators in the class of univalent functions,” Fractional Calculus and Applied Analysis, vol. 9, pp. 159–176, 2006. View at: Google Scholar
12. H. M. Srivastava, R. Agarwal, and S. Jain, “Integral transform and fractional derivative formulas involving the extended generalized hypergeometric functions and probability distributions,” Mathematical Methods in the Applied Sciences, vol. 40, no. 1, pp. 255–273, 2017. View at: Publisher Site | Google Scholar
13. R. Srivastava, R. Agarwal, and S. Jain, “A family of the incomplete hypergeometric functions and associated integral transform and fractional derivative formulas,” Filomat, vol. 31, no. 1, pp. 125–140, 2017. View at: Publisher Site | Google Scholar
14. P. Agarwal and C. L. Koul, “On generating functions,” Journal of Rajasthan Academy of Physical Sciences, vol. 2, no. 3, pp. 173–180, 2003. View at: Google Scholar
15. P. Agarwal, M. Chand, and S. D. Purohit, “A note on generating functions involving the generalized Gauss hypergeometric functions,” National Academy Science Letters, vol. 37, no. 5, pp. 457–459, 2014. View at: Publisher Site | Google Scholar
16. P. Agarwal, M. Chand, and G. Singh, “On new sequence of functions involving the product of .,” Communications in Numerical Analysis, vol. 2015, no. 2, pp. 139–148, 2015. View at: Publisher Site | Google Scholar
17. M. Chand, P. Agarwal, and J. Choi, “Note on generating relations associated with the generalized Gauss hypergeometric function,” Applied Mathematical Sciences, vol. 10, no. 35, pp. 1747–1754, 2016. View at: Publisher Site | Google Scholar
18. 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
19. J. Choi and P. Agarwal, “Certain class of generating functions for the incomplete hypergeometric functions,” Abstract and Applied Analysis, vol. 2014, Article ID 714560, 5 pages, 2014. View at: Publisher Site | Google Scholar
20. H. M. Srivastava, “Certain generating functions of several variables,” ZAMM—Zeitschrift für Angewandte Mathematik und Mechanik, vol. 57, no. 6, pp. 339-340, 1977. View at: Publisher Site | Google Scholar
21. H. M. Srivastava, “Generating relations and other results associated with some families of the extended Hurwitz-Lerch Zeta functions,” SpringerPlus, vol. 2, no. 67, pp. 1–14, 2013. View at: Publisher Site | Google Scholar
22. H. M. Srivastava, “A new family of the -generalized Hurwitz-Lerch Zeta functions with applications,” Applied Mathematics & Information Sciences, vol. 8, no. 4, pp. 1485–1500, 2014. View at: Publisher Site | Google Scholar
23. H. M. Srivastava and P. W. Karlsson, “Multiple Gaussian hypergeometric series,” Ellis Horwood Series: Mathematics and Its Applications, Ellis Horwood Ltd., Chichester, UK, 1985. View at: Google Scholar
24. H. M. Srivastava and H. L. Manocha, A Treatise on Generating Functions, Ellis Horwood Ltd., Chichester, UK, 1984.
25. H. M. Srivastava, M. A. Özarslan, and C. Kaanoğlu, “Some families of generating functions for a certain class of three-variable polynomials,” Integral Transforms and Special Functions, vol. 21, no. 12, pp. 885–896, 2010. View at: Publisher Site | Google Scholar
26. H. M. Srivastava and J. Choi, Zeta and q-Zeta Functions and Associated Series and Integrals, Elsevier Science Publishers, Amsterdam, Netherlands, 2012.

Copyright © 2020 Shilpi Jain 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.