Search

We study the maximum value of the confluent hypergeometric function with oscillatory conditions of parameters. As a consequence, we obtain new inequalities for the Gauss hypergeometric function.

In this paper, we derive some Fourier transforms of confluent hypergeometric functions. We give generalizations of several well-known results involving Fourier transforms of gamma functions. In particular, the generalizations include some Ramanujan's remarkable formulas. [ABSTRACT FROM AUTHOR]

We study the maximum value of the confluent hypergeometric function with oscillatory conditions of parameters. As a consequence, we obtain new inequalities for the Gauss hypergeometric function. [ABSTRACT FROM AUTHOR]

Let be the Jacobi measure supported on the interval [-1, 1]. Let us introduce the Sobolev-type inner product , where . In this paper we prove a Cohen-type inequality for the Fourier expansion in terms of the orthonormal polynomials associated with the above Sobolev inner product. We follow Dreseler and Soardi (1982) and Markett (1983) papers, where such inequalities were proved for classical orthogonal expansions.

In this paper, the partial fractions expansions for the generalized hypergeometric function will be presented. Our results generalized several well-known results in the literature. [ABSTRACT FROM AUTHOR]

In this paper the ladder operator approach has been applied to deduce pointwise asymptotics of the ratio orthogonal polynomials at the endpoints and at the algebraic singularity of the orthogonality weightwhere α, β, γ>0,x0∈(−1, 1), and withhreal analytic and strictly positive on [−1, 1]. In particular, when α=β,x0=0, andheven real analytic and strictly positive on [−1, 1], we obtain the Mehler–Heine formula for corresponding Gegenbauer-type polynomials. [ABSTRACT FROM PUBLISHER]

Let (Q(α,β)n (x)) n⩾0 denote the sequence of polynomials orthogonal with respect to the non-discrete Sobolev inner product = ∫1 -1 f(x)g(x)dμα,β(x) + λ ∫1 -1 f'(x)g'(x)dμα+1,β(x), where λ > 0 and dμα,β(x) = (1 - x)α(1 + x)βdx with α > -1, β > -1. In this paper, we prove a Cohen type inequality for the Fourier expansion in terms of the orthogonal polynomials (Q (α,β)n (x)) n. Necessary conditions for the norm convergence of such a Fourier expansion are given. Finally, the failure of almost everywhere convergence of the Fourier expansion of a function in terms of the orthogonal polynomials associated with the above Sobolev inner product is proved. [ABSTRACT FROM AUTHOR]