Related papers: Multi-integral representations for associated Lege…
In this paper we study the Hilbert scales defined by the associated Legendre functions for arbitrary integer values of the parameter. This problem is equivalent to study the left-definite spectral theory associated to the modified Legendre…
There are 13 equivalence classes of 2D second order quantum and classical superintegrable systems with nontrivial potential, each associated with a quadratic algebra of hidden symmetries. We study the finite and infinite irreducible…
In this paper, we present several novel integral representations of Catalan's constant. We begin by deriving an initial result expressed as a double integral. Subsequently, as a consequence of this result, we establish a general theorem…
Explicit solutions of differential equations of complex fractional orders with respect to functions and with continuous variable coefficients are established. The representations of solutions are given in terms of some convergent infinite…
Looking for a quantum field theory model of Archimedean algebraic geometry a class of infinite-dimensional integral representations of classical special functions was introduced. Precisely the special functions such as Whittaker functions…
To a function with values in the power set of a pre--ordered, separated locally convex space a family of scalarizations is given which completely characterizes the original function. A concept of a Legendre--Fenchel conjugate for set-valued…
Generating functions and functional equations of Dickson polynomials of the first and second kind are derived and continued analytically. These formulae are expressed in terms of the incomplete gamma function over complex variables of the…
In this note, we derive a finite summation formula and an infinite summation formula involving Harmonic numbers of order up to some order by means of several definite integrals
We define and study multivariate exponential functions, symmetric with respect to the alternating group A_n, which is a subgroup of the permutation (symmetric) group S_n. These functions are connected with multivariate exponential…
In this work at first the relation the Mittag-Lefler function to the exponential is given. The results are applied to the construction of the solution of Cauchy problem for ordinary linear operator differential equations with constant…
It is classical that univariate algebraic functions satisfy linear differential equations with polynomial coefficients. Linear recurrences follow for the coefficients of their power series expansions. We show that the linear differential…
A correlation function of the classical orthogonal polynomials is defined and determined. The correlation function obeys a second order difference equation in two variables. The correlation function for the Gegenbauer, Chebyshev and…
First order formulas in a relational signature can be considered as operations on the relations of an underlying set, giving rise to multisorted algebras we call first order algebras. We present universal axioms so that an algebra satisfies…
We formulate for function fields an analog of Serre's conjecture on the modularity of 2-dimensional irreducible mod l representations of the absolute Galois group of Q: our analog is not restricted to 2-dimensional represntations. While the…
We establish effective mean-value estimates for a wide class of multiplicative arithmetic functions, thereby providing (essentially optimal) quantitative versions of Wirsing's classical estimates and extending those of Hal\'asz. Several…
Using a special case of the Efros theorem which was derived by Wlodarski, and operational calculus, it was possible to derive many infinite integrals, finite integrals and integral identities for the function represented by the inverse…
In this paper, we introduce two new non-singular kernel fractional derivatives and present a class of other fractional derivatives derived from the new formulations. We present some important results of uniformly convergent sequences of…
We show that any multiple-valued function can be represented by a linear lambda term typed in a second-order polymorphic type system, using two distinct styles. The first is a circuit style, which mimics combinational circuits in switching…
We present explicit expressions for multi-fold logarithmic integrals that are equivalent to sums over polygamma functions at integer argument. Such relations find application in perturbative quantum field theory, quantum chemistry, analytic…
An umbral type formalism is used to derive integrals involving products of Laguerre polynomials and other special functions.