The Ramanujan Journal最新文献

We study meromorphic modular forms associated with positive definite binary quadratic forms and their cycle integrals along closed geodesics in the modular curve. We show that suitable linear combinations of these meromorphic modular forms have rational cycle integrals. Along the way we evaluate the cycle integrals of the Siegel theta function associated with an even lattice of signature (1, 2) in terms of Hecke’s indefinite theta functions.

In an earlier work, a method was introduced for obtaining indefinite q-integrals of q-special functions from the second-order linear q-difference equations that define them. In this paper, we reformulate the method in terms of q-Riccati equations, which are nonlinear and first order. We derive q-integrals using fragments of these Riccati equations, and here only two specific fragment types are examined in detail. The results presented here are for the q-Airy function, the Ramanujan function, the discrete q-Hermite I and II polynomials, the q-hypergeometric functions, the q-Laguerre polynomials, the Stieltjes-Wigert polynomial, the little q-Legendre and the big q-Legendre polynomials.

We investigate some Appell-type orthogonal polynomial sequences on q-quadratic lattices and we provide some entirely new characterizations of some special cases of the Al-Salam–Chihara polynomials (including the Rogers q-Hermite polynomials). The corresponding regular forms are well described. We also show that the Rogers q-Hermite polynomials constitute a nice orthogonal polynomial base to use when dealing with problems related with the Askey-Wilson and the averaging operators. The proposed method can be applied to similar and to more general problems involving the Askey-Wilson and the Averaging operators, in order to obtain new characterization theorems for classical and semiclassical orthogonal polynomials on lattices.

In this article, we extend the classical framework for computing discriminants of special quasi-orthogonal polynomials from Schur’s resultant formula, and establish a framework for computing discriminants of a sufficiently broader class of polynomials from the resultant formulas that are proven by Ulas and Turaj. More precisely, we derive a formula for the discriminant of a sequence ({r_{A,n}+c r_{A,n-1}}) of polynomials. Here, c is an element of a field K and ({r_{A,n}}) is a sequence of polynomials satisfying a certain recurrence relation. There are several works computing the discriminants of given polynomials. For example, Kaneko–Niiho and Mahlburg–Ono independently proved the formula for the discriminants of certain hypergeometric polynomials that are related to j-invariants of supersingular elliptic curves. Sawa–Uchida proved the formula for the discriminants of quasi-Jacobi polynomials and applied it to prove the nonexistence of certain rational quadrature formulas. Our main theorem presents a uniform way to prove a vast generalization of the above formulas for the discriminants.

We discuss properties of the Hardy-Ramanujan taxicab number, 1729, and similar numbers. The similar numbers include Carmichael numbers, Lucas Carmichael numbers, sums of cubes, and integers of the form (b^npm 1).

Let (R_{(1, 1)}(n)) denote the coefficients of the Dirichlet series (zeta '(s) L'(s, chi _{4})= Sigma _{n= 1}^{infty } R_{(1, 1)}(n) n^{- s}) for (Re s> 1) and (P_{(1)} (x)) the error term of (Sigma _{nle x} R_{(1, 1)}(n).) A representation of the Chowla–Walum type formula for (P_{(1)}(x)) is derived. As a direct application, we shall give a new order estimate for (P_{(1)}(x)), which constitutes an improvement over the evaluation originating from Furuya et al. Furthermore, the asymptotic formula of the integral (int _{1}^{X} P_{(1)}^{k}(x) d x) is established for (k=3, 4).

We study partitions of complex numbers as sums of non-negative powers of a fixed algebraic number (beta ). We prove that if ( beta ) is real quadratic, then the number of partitions is always finite if and only if some conjugate of (beta ) is larger than 1. Further, we show that for (beta ) satisfying a certain condition, the partition function attains all non-negative integers as values.