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Abstract: The aim of this paper is to deal with applications of umbral calculus on fermionic p-adic integral on . From those applications, we derive some new identities on Genocchi numbers and polynomials. Moreover, a systemic study of the class of Sheffer sequences on the generating function of Genocchi polynomials are given. [Copyright &y& Elsevier]

We study combinatorial and number-theoretic issues arising from group-invariant CR mappings from spheres to hyperquadrics. Given an initial complex analytic function, we define two sequences of complex numbers. We derive formulae for these sequences in terms of the Taylor coefficients of the initial function and, in the polynomial case, also in terms of the reciprocals of the roots. We connect these formulae to CR mappings invariant under arbitrary representations of finite cyclic subgroups of the unitary group. We prove that the resultant of them-th polynomial (which in all cases has integer coefficients) and any of its partial derivatives is divisible bymm. [ABSTRACT FROM AUTHOR]

In this same journal, Coons published recently a paper [The transcendence of series related to Stern's diatomic sequence, Int. J. Number Theory6 (2010) 211-217] on the function theoretical transcendence of the generating function of the Stern sequence, and the transcendence over ℚ of the function values at all non-zero algebraic points of the unit disk. The main aim of our paper is to prove the algebraic independence over ℚ of the values of this function and all its derivatives at the same points. The basic analytic ingredient of the proof is the hypertranscendence of the function to be shown before. Another main result concerns the generating function of the Stern polynomials. Whereas the function theoretical transcendence of this function of two variables was already shown by Coons, we prove that, for every pair of non-zero algebraic points in the unit disk, the function value either vanishes or is transcendental. [ABSTRACT FROM AUTHOR]