Journal of Mathematical Analysis ISSN: 2217-3412, URL: www.ilirias.com/jma Volume 8 Issue 2 (2017), Pages 64-72. ON THE VOLKENBORN INTEGRAL OF THE q-EXTENSION OF THE p-ADIC GAMMA FUNCTION ¨ OZGE C ¸ OLAKO ˘ GLU HAVARE, HAMZA MENKEN Abstract. In the present work we consider the q-extension of the p-adic gamma function. We derive the Volkenborn integral of the q-extension of the p-adic gamma function by using its Mahler expansion. Moreover, we give a new representation for the q-extension of the p-adic Euler constant. 1. Introduction The q-Calculus appeared in the 18th century and it continues to develop rapidly. The q-calculus has a great interest and has been studied by Euler, Gauss who discovered q-binomial formula and others. The systematic development of the q- calculus began with FH Jackson in the early 20th century. Although the q-calculus has been studied for over a century, q-analogue of special numbers and polynomials are still of interest [1]. The letter q stands for ‘quantum’, and the q-binomial coefficients play an impor- tant role in ‘quantum calculus’ similar to that of the ordinary binomial coefficients in ordinary calculus. Also, the binomial coefficients are also known combinations or combinatorial numbers. In constructing the properties and identities of some special numbers, binomial coefficients have great interest. The q-analogue of the binomial coefficients an important role play in developing the theory of the q-analogue of these special numbers [5]. The q-binomial coefficients or Gaussian polynomials appear in many identities on q-series. In addition, they are studied in several combinatorial environments as partitions of integers. For an easy handling of the q-binomial coefficients in combinatory it is essential to be familiar with the basic combinatorial structures that admit those coefficients as generating polynomials [9]. The p−adic numbers introduced by the German mathematician Kurt Hensel (1861–1941), are widely used in mathematics: in number theory, algebraic geome- try, representation theory, algebraic and arithmetical dynamics, and cryptography. The p−adic numbers have been used applying fields with successfully applying in 2000 Mathematics Subject Classification. 11S80,11E95, 11S23. Key words and phrases. q-extension of p-adic gamma function; Volkenborn integral; q- extension of p-adic Euler constant; q-binomial coefficient. c 2017 Ilirias Research Institute, Prishtin¨ e, Kosov¨ e. Submitted November 25, 2016. Published April 1, 2017. Communicated by Farrukh Mukhamedov. 64