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The main purpose of this paper is to introduce and investigate a class of generalized Bernoulli polynomials and Euler polynomials based on the generating function. we unify all forms of q-exponential functions by one more parameter. we…

We consider a wide class of determinants whose entries are moments of the so-called semiclassical functionals and we show that they are tau functions for an appropriate isomonodromic family which depends on the parameters of the symbols for…

We compute Fourier transforms of functions expressed as a ratio of one of the Jacobi elliptic functions divided by $\sinh(\pi x)$ or $\cosh(\pi x)$. In many cases, the resulting Fourier transform remains within the same class of functions.…

We explore a number of functional properties of the $q$-gamma function and a class of its quotients; including the $q$-beta function. We obtain formulas for all higher logarithmic derivatives of these quotients and give precise conditions…

We look at the functional Y(M) = int_M K(x) dV(x) for compact Riemannian 2d-manifolds M, where K(x) = (2d)! (d!)^-1 (4pi)^-d int_T prod_k=1^d K_t_2k,t_2k+1(x) dt involves products of d sectional curvatures K_ij(x) averaged over the space T…

We consider the $1$- and $2$-d bicomplex analogs of the classical Fourier--Wigner transform. Their basic properties, including Moyal's identity and characterization of their ranges giving rise to new bicomplex--polyanalytic functional…

For odd square-free n > 1 the n-th cyclotomic polynomial satisfies an identity of Gauss. There are similar identity of Aurifeuille, Le Lasseur and Lucas. These identities all involve certain polynomials with integer coefficients. We show…

The theoretical computing of special values assumed by the hypergeometric functions has a high interest not only on its own, but also in sight of the remarkable implications to both pure Mathematics and Mathematical Physics. Accordingly, in…

We study theta functions of a Riemann surface of genus g from the view point of tau function of a hierarchy of soliton equations. We study two kinds of series expansions. One is the Taylor expansion at any point of the theta divisor. We…

We compute the generating function for the characters of the irreducible representations of SU(n) whose associated Young diagrams have only two rows with the same number of boxes. The result is a rational determinantal expression in which…

We write the Euler characteristic X(G) of a four dimensional finite simple geometric graph G=(V,E) in terms of the Euler characteristic X(G(w)) of two-dimensional geometric subgraphs G(w). The Euler curvature K(x) of a four dimensional…

We introduce a natural definition for sums of the form \[ \sum_{\nu=1}^x f(\nu) \] when the number of terms x is a rather arbitrary real or even complex number. The resulting theory includes the known interpolation of the factorial by the…

The d'Arcais polynomials $P_n(z)$ for $n\in\{0,1,\dots\}$ are defined as $\sum_{n=0}^{\infty} P_n(z) q^n = \exp(-z\ln((q;q)_{\infty}))$ where the $q$-Pochhammer symbol is $(q;q)_{\infty} = \prod_{k=1}^{\infty} (1-q^k)$ for $|q|<1$. Denoting…

We consider solvable matrix models. We generalize Harish-Chandra-Itzykson-Zuber and certain other integrals (Gross-Witten integral and integrals over complex matrices) using the notion of tau function of matrix argument. In this case one…

Let $1=d_{1}<d_{2}< \cdots < d_{\tau(n)}=n$ denote the ordered sequence of the positive divisors of an integer $n$. We are interested in estimating the arithmetic function $$ V(n) := \prod_{1 \le i < j \le \tau(n)}(d_{j}-d_{i}) \quad (n \ge…

Double ramification loci parametrise marked curves where a weighted sum of the markings is linearly trivial; higher-rank loci are obtained by imposing several such conditions simultaneously. We obtain closed formulae for the orbifold Euler…

In the paper, the author elementarily unifies and generalizes eight identities involving the functions $\frac{\pm1}{e^{\pm t}-1}$ and their derivatives. By one of these identities, the author establishes two explicit formulae for computing…

There are (at least) two different approaches to define equivariant analogue of the Euler charateristic for a space with a finite group action. The first one defines it as an element of the Burnside ring of the group. The second approach…

We find a formula that relates the Fourier transform of a radial function on $\mathbf{R}^n$ with the Fourier transform of the same function defined on $\mathbf{R}^{n+2}$. This formula enables one to explicitly calculate the Fourier…

We investigate the existence of Boolean degree $d$ functions on the Grassmann graph of $k$-spaces in the vector space $\mathbb{F}_q^n$. For $d=1$ several non-existence and classification results are known, and no non-trivial examples are…