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In this paper, we deal with the zeros of the q-shift difference-differential polynomials [ P ( f ) ∏ j = 1 d f ( q j z + c j ) s j ] ( k ) − α ( z ) and ( P ( f ) ∏ j = 1 d [ f ( q j z + c j ) − f ( z ) ] s j ) ( k ) − α ( z ) , where P ( f ) is a nonzero polynomial of degree n, q j , c j ∈ C ∖ { 0 } ( j = 1 , … , d ) are constants, n , d , s j ( j = 1 , … , d ) ∈ N + and α ( z ) is a small function of f. The results of this paper are an extension of the previous theorems given by Chen and Chen and Qi. We also investigate the value sharing for q-shift difference polynomials of entire functions and obtain some results which extend the recent theorem given by Liu, Liu and Cao. MSC:39A50, 30D35.

In this paper, we consider Barnes-type special polynomials and give some identities of their polynomials which are derived from the bosonic p-adic integral or the fermionic p-adic integral on $\mathbb{Z}_{p}$ .

In this paper, the author puts forward a kind of anti-periodic boundary value problems of fractional equations with the Riemann-Liouville fractional derivative. More precisely, the author is concerned with the following fractional equation: with the anti-periodic boundary value conditions where denotes the standard Riemann-Liouville fractional derivative of order , and the nonlinear function may be singular at . By applying the contraction mapping principle and the other fixed point theorem, the author obtains the existence and uniqueness of solutions. MSC:34A08, 34B15.

In this paper, we investigate the growth of some functions that share functions with their derivative of higher order. The first main theorem is an improvement of the result obtained by Lü (Bull. Korean Math. Soc. 48: 951-957, 2011), two examples are given to show that the conclusion is sharp. The second main theorem is of estimating, more exactly, the order of an entire function sharing polynomial, which extends the related result of Lü, Xu and Chen (Arch. Math. 92: 593-601, 2009). MSC:30D35, 30D45.

In the current paper, we study the stability and the superstability of ∗-derivations associated with the Cauchy functional equation and the Jensen functional equation. We also prove the stability and the superstability of Jordan ∗-derivations on Banach ∗-algebras.

Motivated by the recent investigations of several authors, in this paper, we derive several new expansion formulas involving a generalized Hurwitz-Lerch zeta function introduced and studied recently by Srivastavaet al. (Integral Transforms Spec. Funct. 22:487-506, 2011). These expansions are obtained by using some fractional calculus theorems such as the generalized Leibniz rules for the fractional derivatives and the Taylor-like expansions in terms of different functions. Several (known or new) special cases are also considered.MSC:11M25, 11M35, 26A33, 33C05, 33C60.

This paper investigates the asymptotic behavior of solutions of the mixed type neutral differential equation with impulsive perturbations , , , , . Sufficient conditions are obtained to guarantee that every solution tends to a constant as . Examples illustrating the abstract results are also presented. MSC:34K25, 34K45.

İkikardeş, Nazlı Yıldız (Balikesir Author), In this paper, we investigate the convolution sums Sigma((a+b+c)x=n) a, Sigma(ax+by=n) ab, Sigma((ax+by+cz)x=n) abc, Sigma((ax+by+cz+du)x=n) abcd, where a, b, c, d, x, y, z, u, n is an element of N. Many new equalities and inequalities involving convolution sums, Bernoulli numbers and divisor functions have also been given., Korean Government - B21303 National Institute for Mathematical Sciences (NIMS)

In this paper, we investigate the growth of meromorphic solutions of the equations where (≢0), and (≢0) are entire functions of finite order. We find some conditions on the coefficients to guarantee that every nontrivial meromorphic solution of such equations is of infinite order. MSC:30D35, 34M10.

In this paper, we investigate the family of difference Painleve IV equation , where R is rational in w and meromorphic in z. If the equation assumes an admissible meromorphic solution of hyper-order , we fix the degree of , and we give some further discussions. MSC:39A10.

In this paper, we prove some results on the uniqueness of meromorphic functions sharing small functions CM with their linear difference polynomials. Examples are provided to show the existence of meromorphic functions satisfying the conditions of our results.

The main object of the present paper is to show certain sufficient conditions for univalency of analytic functions with missing coefficients.

In this paper, we introduce the notion of generalized Fibonacci sequences over a groupoid and discuss it in particular for the case where the groupoid contains idempotents and pre-idempotents. Using the notion of Smarandache-type P-algebra, we obtain several relations on groupoids which are derived from generalized Fibonacci sequences. MSC:11B39, 20N02, 06F35.