On Generating Relations Associated with the Extended Gauss and Confluent Hypergeometric Functions


Download PDF

Authors: D. SRIVASTAVA, M. GHAYASUDDIN AND W. A. KHAN

DOI: 10.46793/KgJMat2609.1531S

Abstract:

In this research note, we establish a new class of generating relations associated with the extended Gauss and confluent hypergeometric functions using the concept of Hadamard product. Some deductions of our main results are also indicated.



Keywords:

Gauss hypergeometric function, extended Gauss hypergeometric function, confluent hypergeometric function, extended confluent hypergeometric function, Hadamard product.



References:

[1]   D. M. Lee, A. K. Rathie, R. K. Parmar and Y. S. Kim, Generalization of extended beta function, hypergeometric and confluent hypergeometric functions, Honam Math. J. 33(2011), 187–206.

[2]   E. Özergin, M. A. Özarsalan and A. Altin, Extension of gamma, beta and hypergeometric functions, J. Comput. Appl. Math. 235 (2011), 4601–4610.

[3]   H. M. Srivastava, P. Agarwal and S. Jain, Generating functions for the generalized Gauss hypergeometric functions, Appl. Math. Comput. 247 (2014), 348–352.

[4]   H. M. Srivastava and H. L. Manocha, A Treatise on Generating Functions, Halsted Press (Ellis Horwood Limited, Chichester), John Wiley and Sons, New York, 1984.

[5]   J. Choi, A. K. Rathie and R. K. Parmar, Extension of extended beta, hypergeometric and confluent hypergeometric functions, Honam Math. J. 36 (2014), 357–385. http://dx.doi.org/10.5831/HMJ.2014.36.2.357

[6]   J. Choi, M. Ghayasuddin and N. U. Khan, Generalized extended Whittaker function and it’s properties, App. Math. Sci. 9 (2015), 6529–6541. https://doi.org/10.12988/ams.2015.58555

[7]   M. Ghayasuddin and N. U. Khan, Some transformations and generating relations of multivariable hypergeometric functions, Palest. J. Math. 6(2) (2017), 165–173.

[8]   M. A. Chaudhary, A. Qadir, H. M. Srivastava and R. B. Paris, Extended hypergeometric and confluent hypergeometric functions, Appl. Math. Comput. 159 (2004), 589–602. https://doi.org/10.1016/j.amc.2003.09.017

[9]   M. Chand, P. Agarwal and J. Choi, Note on generation relations associated with the generalized Gauss hypergeometric function, Appl. Math. Sci. 10(35) (2016), 1747–1754. https://doi.org/10.1007/s40009-014-0250-7

[10]   N. U. Khan and M. Ghayasuddin, Generalization of extended Appell’s and Lauricella’s hypergeometric functions, Honam Math. J. 37(1) (2015), 113–126. https://doi.org/10.5831/HMJ.2015.37.1.113

[11]   N. U. Khan and M. Ghayasuddin, A note on generalized extended Whittaker function, Honam Math. J. 38(2) (2016), 325–335. https://doi.org/10.5831/HMJ.2016.38.2.325

[12]   N. U. Khan, T. Usman and M. Ghayasuddin, A new generalization of confluent hypergeometric function and Whittaker function, Bol. Soc. Parana. Mat. 38(2) (2020), 9–26. https://doi.org/10.5269/bspm.v38i2.37578

[13]   P. Agarwal, M. Chand and S. D. Purohit, A note on generating functions involving the generalized Gauss hypergeometric function, National Acad. Sci. Lett. 37 (2014), 457–459.

[14]   P. Agarwal and C. L. Koul, On generating functions, J. Rajasthan Acad. Phy. Sci. 2 (2003), 173–180.

[15]   R. K. Parmar, A new generalization of gamma, beta, hypergeometric and confluent hypergeometric functions, Le Matematiche LXVIII (2013), 33–52.

[16]   T. Pohlen, The Hadamard product and universal power series, Ph.D. Thesis, University Trier, 2009.

[17]   M. Ali, A further extensions of beta and related functions, Palest. J. Math. 12(3) (2023), 65–73.

[18]   S. Ali, N. K. Regar and S. Parida, On generalized extended beta and hypergeometric functions, Honam Math. J. 46(2) (2024), 313–334. https://doi.org/10.5831/HMJ.2024.46.2.313

[19]   R. K. Parmar and T. K. Pogany, Bounds for novel extended beta and hypergeometric functions and related results, J. Inequal. Appl. 2024(77) (2024). https://doi.org/10.1186/s13660-024-03148-8