Abstract
This paper deals with the study of the zeros of the big q-Bessel functions. In particular, we prove new orthogonality relations for functions which are similar to the one for the classical Bessel functions. Also we give some applications related to the sampling theory.
Highlights
The classical Bessel functions Jα ( x ) which are defined by [1] ( x/2)α Jα ( x ) := Γ ( α + 1) 1 x2 − ;− α+1 !, satisfy the orthogonality relations Z 1
Other q-analogues can be obtained as formal limits of the three q-analogues of Jacobi polynomials; i.e., of little q-Jacobi polynomials, big q-Jacobi polynomials and Askey-Wilson polynomials. For this reason we propose to speak about little q-Bessel functions, big q-Bessel functions and AW type q-Bessel functions for the corresponding limit cases
In this work we show that all zeros of the big q-Bessel function Jα ( x, λ; q2 ) are real and simple
Summary
Where { jαn }n∈N are the zeros of Jα ( x ). Other q-analogues can be obtained as formal limits of the three q-analogues of Jacobi polynomials; i.e., of little q-Jacobi polynomials, big q-Jacobi polynomials and Askey-Wilson polynomials. In this paper we discuss a new orthogonality relations for the big q-Bessel functions [6]. In this work we show that all zeros of the big q-Bessel function Jα ( x, λ; q2 ) are real and simple. The classical sampling theorem of Whittaker-Kotelnikov-Shannon (WKS), states that band-limited functions can be recovered from their values at the integers. In this work we provide a q-version of the sampling theorem of Whittaker-Kotelnikov-Shannon, and q-type band-limited signals which are defined in terms of Jackson’s q-integral. In the last section, we give a q-version of the sampling theorem in the points jnα
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