COSINE HIGHER-ORDER EULER NUMBER CONGRUENCES AND DIRICHLET <i>L</i>-FUNCTION VALUES

Abstract

Abstract. In this paper we obtain the residue modulo a prime power of cosine higher-order Euler numbers H (k) 2n (m) in terms of the linear combination of the Dirichlet L-function values L(s, χ) at positive integral arguments s or of generalized Bernoulli numbers. Our results are restricted to the equal parity case; i.e. s and χ are of the same parity. In the process, we employ Yamamoto's results on finite expressions in terms of Dirichlet L-function values for short interval character sums and in this sense our treatment is decisive, i.e. any ad-hoc transformation of short interval sums. The results obtained not only generalize the previous results pertaining to the congruences modulo a prime power of the class numbers as the special case of s = 1 in terms of Euler numbers but also closes the chapter on possible similar research.

Various Structures of the Roots and Explicit Properties of q-cosine Bernoulli Polynomials and q-sine Bernoulli Polynomials

“…Based on the above, many studies can confirm various polynomials and their properties (see [28]). The main aim of this paper is to identify the property of q-cosine Bernoulli polynomials and q-sine Bernoulli polynomials.…”

“…Acknowledging their significance, many mathematicians are familiar with these numbers and polynomials, and they have been studied for a long time. The previous definitions and theorems are also applied to polynomials, and their properties are studied in various ways in combination with Bernoulli, Euler, and Genocchi polynomials, which are considered important (see [1,5,[13][14][15][16][17][18][20][21][22][23][24][25][26]28]). The definition of q-Bernoulli polynomials is as follows: Definition 7. q-Bernoulli numbers, B n,q , and polynomials, B n,q (z), can be expressed as(see [17]):…”