Some $k$-Horn hypergeometric functions and their properties

Year 2023, Volume: 12 Issue: 2, 97 - 107, 31.08.2023

Caner Çatak Recep Şahin Ali Olgun Oğuz Yağcı

https://doi.org/10.54187/jnrs.1335407

Abstract

In the theory of special functions, the $k$-Pochhammer symbol is a generalization of the Pochhammer symbol. With the help of the $k$-Pochhammer symbol, we introduce and study a new generalization of the $k$-Horn hypergeometric functions such as, ${G}_{1}^{k}$, ${G}_{2}^{k}$ and ${G}_{3}^{k}$. Furthermore, several investigations have been carried out for some important recursion formulae for several one variable and two variables $k$-hypergeometric functions. In the light of these studies, we introduce some important recursion formulae for several newly generalized $k$-Horn hypergeometric functions. Finally, we present several relations between some $k$-Horn hypergeometric functions ${G}_{1}^{k}$, ${G}_{2}^{k}$ and ${G}_{3}^{k}$, and $k$-Gauss hypergeometric functions $_{2}{F}_{1}^{k}$.

Keywords

$k$-Gamma function, $k$-Hypergeometric function, $k$-Gauss hypergeometric function, Recursion formula

References

• P. Agarwal, M. Chand, J. Choi, Some integrals involving-functions and Laguerre polynomials, Ukrainian Mathematical Journal 71 (9) (2020) 1321-1340.
• E. Ata, M-Lauricella hypergeometric functions: integral representations and solutions of fractional differential equations, Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 72 (2) (2023) 512-529.
• J. Choi, M. I. Qureshi, A. H. Bhat, J. Majid, Reduction formulas for generalized hypergeometric series associated with new sequences and applications, Fractal and Fractional 5 (4) (2021) Article Number 150 23 pages.
• H. M. Srivastava, A. Çetinkaya, İ. O. Kıymaz, A certain generalized Pochhammer symbol and its applications to hypergeometric functions, Applied Mathematics and Computation 226 (2014) 484-491.
• R. Şahin, O. Yağcı, A new generalization of Pochhammer symbol and its applications, Applied Mathematics and Nonlinear Sciences 5 (1) (2020) 255-266.
• R. Şahin, O. Yağcı, Fractional calculus of the extended hypergeometric function, Applied Mathematics and Nonlinear Sciences 5 (1) (2020) 369-384.
• Y. A. Brychkov, Handbook of special functions, derivatives, integrals, series and other formulas, CRC Press, Boca Raton, 2008.
• B. Davies, Integral transforms and their applications, New York, Springer-Verlag, 1984.
• G. Lohöfer, Theory of an electromagnetically deviated metal sphere. I: absorbed power, SIAM Journal on Applied Mathematics 49 (2) (1989) 567-581.
• A. M. Mathai, R. K. Saxena, Generalized hypergeometric functions with applications in statistics and physical sciences, Springer-Verlag, Berlin, Heidelberg and New York, 1973.
• H. M. Srivastava, J. Choi, Zeta and $q$-Zeta functions and associated series and integrals, Elsevier Science Publishers, Amsterdam, London and New York, 2012.
• H. M. Srivastava, P. W. Karlsson, Multiple Gaussian hypergeometric series, Ellis Horwood, Chichester, 1984.
• H. M. Srivastava, B. R. K. Kashyap, Special functions in queuing theory and related stochastic processes, Academic Prees, New York, London and San Francisco, 1982.
• R. Díaz, E. Pariguan, On hypergeometric functions and Pochhammer, Divulgaciones Matemáticas 15 (2) (2007) 179-192.
• Ö. G. Yılmaz, R. Aktaş, F. Taşdelen, On some formulas for the $k$-analogue of Appell functions and generating relations via $k$-fractional derivative, Fractal and Fractional 4 (4) (2020) Article Number 48 19 pages.
• P. Agarwal, J. Choi, S. Jain, Extended hypergeometric functions of two and three variables, Communication of the Korean Mathematical Society 30 (4) (2015) 403-414.
• J. Choi, Certain applications of generalized Kummer's summation formulas for 2F1, Symmetry 13 (8) (2021) Article Number 1538 20 pages.
• R. Şahin, O. Yağcı, $H_{A}^{(\tau_{1},\tau_{2},\tau_{3})}$ Srivastava hypergeometric function, Mathematical Sciences and Applications E-Notes 6 (2) (2018) 1--9.
• O. Yağcı, $H_{B}^{(\tau_{1},\tau_{2},\tau_{3})}$ Srivastava hypergeometric function, Mathematical Sciences and Applications E-Notes 7 (2) (2019) 195-204.
• O. Yağcı, R. Şahin, Degenerate Pochhammer symbol, degenerate Sumudu transform, and degenerate hypergeometric function with applications, Hacettepe Journal of Mathematics and Statistics 50 (5) (2021) 1-18.
• R. Şahin, S. R. S. Agha, Recursion formulas for $G_{1}$ and $G_{2}$ Horn hypergeometric functions, Miskolc Mathematical Notes 16 (2) (2015) 1153-1162.
• J. A. Younis, New integrals for Horn hypergeometric functions in two variables, Global Journal of Science Frontier Research 20 (6) (2020) 31-40.
• A. Shehata, S. I. Moustafa, On certain new formulas for the Horn's hypergeometric functions $G_{1}$, $G_{2}$ and $G_{3}$, Afrika Matematika 33 (2) (2022) Article Number 65 12 pages.
• A. Shehata, S. I. Moustafa, Some new formulas for Horn's hypergeometric functions $H_{1},\,H_{2},\, H_{3},\, H_{4},\, H_{5},\, H_{6},\, \text{and}\, H_{7}$, Thai Journal of Mathematics 20 (2) (2022) 1011-1030.
• J. Horn, Hypergeometrische funktionen zweier veränderlichen, Mathematische Annalen 105 (1) (1931) 381-407.