A Note on $k$-Gamma Function and Pochhammer $k$-Symbol

Authors

  • Shahid Mubeen Department of Mathematics, University of Sargodha, Sargodha
  • Abdur Rehman Department of Mathematics, University of Sargodha, Sargodha

DOI:

https://doi.org/10.26713/jims.v6i2.252

Keywords:

Factorial function, Pochhammer $k$-symbol, $k$-Gamma function

Abstract

In this note, we discuss some extended results involving the Pochhammer's symbol and express the multiple factorials in terms of the said symbol. We prove the $k$-analogue of Vandermonde's theorem which contains the binomial theorem as a limiting case. Also, we introduce some limit formulae involving the $k$-symbol and prove the $k$-analogue Gauss multiplication and Legendere's duplication theorems by using these formulae.

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References

P. Appell, Sur les series hypergeometriques de deuxvariables, et sur des equations differentielles lineaires aux derivees partielles, C.R Acad.Sci. Paris 90 (1880a), 296-298.

E.D. Rainville, Special Functions,The MacMillan Company, New York, USA (1960).

B.C. Carlson, Special Functions of Applied Mathematics, Academic Press New York San Francisco 197.

K. Kono, Multifactorial, Alien's Mathematics, pp. 1-19.

R. Diaz and E. Pariguan, On hypergeometric functions and $k$-Pochhammer symbol, Divulgaciones Mathematics, 15 (2) (2007), 179-192.

C.G. Kokologiannaki, Properties and inequalities of generalized $k$-gamma, beta and zeta functions, International Journal of Contemp. Math Sciences 5 (14) (2010), 653-660.

C.G. Kokologiannaki and V. Krasniqi, Some properties of $k$-gamma function. LE MATHEMATICS LXVIII (2013), 13-22.

V. Krasniqi, A limit for the $k$-gamma and $k$-beta function, Int. Math. Forum 5(33) (2010), 1613-1617.

M. Mansoor, Determining the $k$-generalized gamma function $Gamma_k(x)$, by functional equations, International Journal Contemp. Math. Sciences 4(21) (2009), 1037-1042.

S. Mubeen and G. M. Habibullah, An integral representation of some $k$-hypergeometric functions, Int. Math. Forum 7(4) (2012), 203-207.

S. Mubeen and G. M. Habibullah, $k$-Fractional integrals and applications, International Journal of Mathematics and Science 7(2) (2012), 89-94.

G. E. Andrews, R. Askey and R. Roy, Special Functions Encyclopedia of mathematics and its Application 71, Cambridge University Press (1999).

S. Mubeen, A. Rehman and F. Shaheen, Properties of $k$-gamma, $k$-beta and $k$-psi functions, Bothalia 44(4) (2014), 372-380.

A.T. Vandermond (1772), Memoire sur des irrationnelles de differens order avec nne application au circle, Histoire Acad. Roy.Sci.acec Mem. Math. Phys, 1772 Printed in Paris, 1775,

pp. 489-498.

R.A. Askey, Orthogonal Polynomials and Special Functions, Reg. Conf. Ser. Appl. Math 21. Soc. Ind. Appl. Math, Philadelphia, Penn Sylvania (1975b).

G. Polya, Problems and Theorems in analysis 1 (1972); 2 (1976), Springer-Veriag, New York.

E.T. Copson, Asymptotic Expansions, Cambridge Univ. Press, London --- New York (1965).

F.W.J. Olver, Asymptotic and Special Functions, Academic Press, New York (1974).

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Published

2014-12-31
CITATION

How to Cite

Mubeen, S., & Rehman, A. (2014). A Note on $k$-Gamma Function and Pochhammer $k$-Symbol. Journal of Informatics and Mathematical Sciences, 6(2), 93–107. https://doi.org/10.26713/jims.v6i2.252

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Research Articles