The oscillation of differential transforms – entire Jacobi expansions

Oscillation of Higher-order Linear Differential Equations with Entire Coefficients

This article is devoted to studying the solutions to the differential equation f (k) +Ak−1(z)f (k−1) + · · ·+A0(z)f = 0, k ≥ 2, where coefficients Aj(z) are entire functions of integer order. We obtain estimates on the orders and the hyper orders of the solutions to the above equation.

Product Jacobi Expansions 3

The coefficients of differentiated expansions of double and triple Jacobi polynomials

Formulae expressing explicitly the coefficients of an expansion of double Jacobi polynomials which has been partially differentiated an arbitrary number of times with respect to its variables in terms of the coefficients of the original expansion are stated and proved. Extension to expansion of triple Jacobi polynomials is given. The results for the special cases of double and triple ultraspher...

On the Fourier expansions of Jacobi forms

We use the relationship between Jacobi forms and vector-valued modular forms to study the Fourier expansions of Jacobi forms of indexes p, p2, and pq for distinct odd primes p, q. Specifically, we show that for such indexes, a Jacobi form is uniquely determined by one of the associated components of the vector-valued modular form. However, in the case of indexes of the form pq or p2, there are ...

Divergent Cesàro Means of Jacobi-Sobolev Expansions

Let μ be the Jacobi measure supported on the interval [−1, 1]. Let introduce the Sobolev-type inner product 〈f, g〉 = ∫ 1 −1 f(x)g(x) dμ(x) +Mf(1)g(1) +Nf ′(1)g′(1), where M,N ≥ 0. In this paper we prove that, for certain indices δ, there are functions whose Cesàro means of order δ in the Fourier expansion in terms of the orthonormal polynomials associated with the above Sobolev inner product ar...