Related papers: A Note on Integrals Containing the Univariate Lomm…
In this paper, we study the formulae for a product of two product Euler polynomials. From this study, we derive some formulae for the integral of the product of two or more Ruler polynomials.
We consider integrals of tau functions of Zakharov-Shabat systems whose higher times are related to the eigenvalues of products of random matrices. Apart of random matrices there is the set of $n$ pairs of given matrices which play the role…
In this work the authors use their contour integral method to derive a double integral connected to the modified Bessel function of the second kind and express it in terms of the Lerch function. There are some useful results relating double…
We obtain definite integrals for products of associated Legendre functions with Bessel functions, associated Legendre functions, and Chebyshev polynomials of the first kind using orthogonality and integral transforms.
The Kronecker product of two Schur functions $s_{\mu}$ and $s_{\nu}$, denoted by $s_{\mu}*s_{\nu}$, is the Frobenius characteristic of the tensor product of the irreducible representations of the symmetric group corresponding to the…
Estimates of some integrals related to variations of smooth functions are presented.
This brief note concerns the invertibility of certain alternant matrices. In particular those that consisting of polynomials and products of polynomials and logarithms are shown to be invertible under appropriate conditions on the degrees…
We consider generalised Mehler semigroups and, assuming the existence of an associated invariant measure $\sigma$, we prove functional integral inequalities with respect to $\sigma$, such as logarithmic Sobolev and Poincar\'{e} type.…
By using Fubini theorem or Tonelli theorem, we find that the zeta function value at 2 is equal to a special integral. Furthermore, We find that this special integral is two times of another special integral. By using this fact we obtain the…
In this paper, we extend Rohrlich's Theorem on the integral of logarithms of meromorphic functions to compute the inner product between such functions and polynomials in the $j$-function. We then show that the generating function for these…
Letting $(t_n)$ denote the Thue-Morse sequence with values $0, 1$, we note that the Woods-Robbins product $$ \prod_{n \geq 0} \left(\frac{2n+1}{2n+2}\right)^{(-1)^{t_n}} = 2^{-1/2} $$ involves a rational function in $n$ and the $\pm 1$…
For one-matrix models with polynomial potentials, the explicit relationship between the partition function and the isomonodromic tau function for the 2x2 polynomial differential systems satisfied by the associated orthogonal polynomials is…
Inspired by recent formul\ae\ of Dubrovin, Yang, and Zagier, we interpret the tau function enumerating stationary Gromov-Witten invariants of $\mathbb{P}^1$ as an isomonodromic tau function associated with a difference equation. As a…
This paper is an exposition of heuristics related to Witten's functional integral, relating it to Vassiliev invariants and to the Kontsevich integrals that can be used to produce Vassiliev invariants of knots and links.In particular, we…
The purpose of this note is to improve the current theoretical results for the correlation functions of the Mobius sequence $\{\mu(n): n\geq 1 \}$ and the Liouville sequence $\{\lambda(n): n\geq 1 \}$.
We investigate a relationship between MacMahon's generalized sum-of-divisors functions and Chebyshev polynomials of the first kind. This determines a recurrence relation to compute these functions, as well as proving a conjecture of…
Theta functions were defined for varieties with effective anticanonical divisor and are related to certain punctured Gromov-Witten invariants. In this paper we show that in the case of a log Calabi-Yau surface (X,D) with smooth very ample…
We introduce a symbolic method for the evaluation of definite integrals containing combinations of various functions, including exponentials, logarithm and products of Bessel functions of different types. The method we develop is naturally…
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…
In this work, considering a general subclass of bi-univalent functions and using the Chebyshev polynomials, we obtain coefficient expansions for functions in this class.