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In the paper, we introduce an analogue of Haar distribution based on ( ρ , q ) -numbers, as follows: μ ρ , q ( a + p N Z p ) = ρ p N [ p N ] ρ , q ( q ρ ) a . By means of this distribution, we derive ( ρ , q ) -analogue of Volkenborn integration which is a new generalization of Kim's q -Volkenborn integration defined in [11] . From this definition, we investigate some properties of Volkenborn integration based on ( ρ , q ) -numbers. Finally, we construct ( ρ , q ) -Bernoulli numbers and polynomials derived from ( ρ , q ) -Volkenborn integral and obtain some their properties. [ABSTRACT FROM AUTHOR]

In this paper, we introduce q -Frobenius–Euler numbers and derive some new properties. From those properties, we show that this number is a p -adic integer, and can be expressed by von Staudt–Clausen's theorem. We also get a type of Kummer congruence for this number. [ABSTRACT FROM AUTHOR]

The Newton power-sum formulas relate to sums of powers of roots of a polynomial with the coefficients of the polynomial. In this paper we obtain formulas that relate to sums of reciprocal powers of zeros and poles of entire and meromorphic functions with the coefficients of their Taylor series expansions. We then derive the recurrence formulas for the Riemann zeta function at integer arguments and compute the sums extended over the nontrivial zeros of the Riemann zeta function. [ABSTRACT FROM AUTHOR]

Abstract: The purpose of this paper is to derive some applications of umbral calculus by using extended fermionic p-adic q-integral on . From those applications, we derive some new interesting properties on the new family of Euler numbers and polynomials. That is, a systemic study of the class of Sheffer sequences in connection with generating function of the weighted q-Euler polynomials is given in the present paper. [Copyright &y& Elsevier]