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We establish the mapping relations between analytic functions and periodic functions using the abstract operators $\cos(h\partial_x)$ and $\sin(h\partial_x)$, including the mapping relations between power series and trigonometric series,…
In this paper, we are interested by the cotangent sum c0(q/p) related to the Estermann zeta function for the special case when q = 1 and get explicit formula for its series expansion, which represents an improvement of the identity (2:1)…
In this article we present certain formulas involving arithmetical functions. In the first part we study properties of sums and product formulas for general type of arithmetic functions. In the second part we apply these formulas to the…
The quantum integer [n]_q is the polynomial 1 + q + q^2 + ... + q^{n-1}, and the sequence of polynomials { [n]_q }_{n=1}^{\infty} is a solution of the functional equation f_{mn}(q) = f_m(q)f_n(q^m). In this paper, semidirect products of…
Various new identities, recurrence relations, integral representations, connection and explicit formulas are established for the Bernoulli, Euler numbers and the values of Riemann's zeta function. To do this, we explore properties of some…
We provide new methods to straightforwardly obtain compact and analytic expressions for epsilon-expansions of functions appearing in both field and string theory amplitudes. An algebraic method is presented to explicitly solve for…
We introduce a novel arithmetic function $w(n)$, a generalization of the Liouville function $\lambda(n)$, as the coefficients of a Dirichlet series. By spatially encoding information in a natural way about the distribution of prime factors…
The celebrated Zeilberger algorithm which finds holonomic recurrence equations for definite sums of hypergeometric terms $F(n,k)$ is extended to certain nonhypergeometric terms. An expression $F(n,k)$ is called a hypergeometric term if both…
Two linear recurrences exhibit mirror symmetry connecting the constants $e$ and $\pi$. When parametrized, their asymptotic connection constants extend to meromorphic functions satisfying additive functional equations with rational…
We prove the equality of several $\tau$-recurrent sequences, which were first considered by Pellarin, and which have close connections to Drinfeld vectorial modular forms. Our result has several consequences: an $A$-expansion for the…
Numerous novel integral and series representations for Ferrers functions of the first kind (associated Legendre functions on the cut) of arbitrary degree and order, various integral, series and differential relations connecting Ferrers…
We study the structure of analytic continuation of solutions of an even rank system of linear ordinary differential equations of Okubo normal form (ONF). We develop an adjustment of the method by using the Euler integral for evaluating the…
We present results for some infinite series appearing in Feynman diagram calculations, many of which are similar to the Euler series. These include both one-dimensional and two-dimensional series. Most of these series can be expressed in…
Let phi denote Euler's phi function. For a fixed odd prime we give an asymptotic series expansion in the sense of Poincare for the number E_q(x) of n<=x such that q does not divide phi(n). Thereby we improve on a recent theorem of B.K.…
The ring of symmetric functions $\Lambda$, with natural basis given by the Schur functions, arise in many different areas of mathematics. For example, as the cohomology ring of the grassmanian, and as the representation ring of the…
Lame equation arises from deriving Laplace equation in ellipsoidal coordinates; in other words, it's called ellipsoidal harmonic equation. Lame functions are applicable to diverse areas such as boundary value problems in ellipsoidal…
The idea of generating integrals analogous to generating functions is first introduced in this paper. A new proof of the well-known Finite Harmonic Series Theorem in Analysis and Analytical Number Theory is then obtained by the method of…
Motivated by recent work of George Andrews and Mircea Merca on the expansion of the quotient of the truncation of Euler's pentagonal number series by the complete series, we provide similar expansion results for averages involving…
This study presents explicit evaluations of the series \begin{equation*} \sum_{k=1}^\infty \frac{H_{k/n}^{(p)}}{k^q} \quad \text{and} \quad \sum_{k=1}^\infty \frac{(-1)^k H_{k/2n}^{(p)}}{k^q}, \quad p,q,n \in \mathbb{Z}_{\ge 1},\; q \ne 1,…
Let $(a;q)_n=\prod_{0\le k<n}(1-aq^k)$ for n=0,1,2,.... Define q-Euler numbers $E_n(q)$, q-Sali\'e numbers $S_n(q)$ and q-Carlitz numbers $C_n(q)$ as follows: $$\sum_{n=0}^{\infty}E_n(q)\frac{x^n}{(q,q)_n}…