Page 23 - Textos de Matemática Vol. 34

CONNECTING BASIC BESSEL FUNCTIONS
LU´IS DANIEL DE ABREU
Dedicated to Professor J. A. Sampaio Martins on the occasion of his sixtieth birthday
Abstract. A brief account on recent developments in the theory of basic Bessel functions is given. Using old formulae due to Euler, new relations be- tween the second and the third Jackson q−Bessel functions are obtained. In the special cases of the q−trigonometric functions, an important simplification occurs.
1. Introduction
During the development of his lifelong research work on q−series, Frederick H.
Jackson introduced three basic analogues (or q−analogues) of the Bessel function
[12], [13]. In the modern notation due to Ismail [10], [11], these q−analogues are (1) (2) (3)
denoted by Jν (z;q), Jν (z;q) and Jν (z;q). If ν > −1, they are defined by the series expansions
(1.1)
(1.2)
and
(1.3)
J(1)(z;q) = ν
J(2)(z; q) = ν
(q; q)∞ (qν+1; q)
z 2n+ν 2
z 2n+ν 2
,
∞
(−1)n
n=0 ∞
qn(n+1) (qν+1; q)n(q; q)n
qn(ν+1) (qν+1; q)n(q; q)n
(qν+1; q) ∞
z2n+ν Jν (z;q)= (q;q) (−1) (qν+1;q) (q;q) 2 ,
∞ (q; q)∞
(−1)n (qν+1;q)∞ n
n=0
∞
n(n+1) q 2
(3)
∞n=0 nn (1.4) (a;q)n =(1−q)(1−aq)...1−aqn−1
2000 Mathematics Subject Classification. 33D15.
Partial financial assistance by Funda¸ca˜o para a Ciˆencia e Tecnologia and Centro de Matem´atica da Universidade de Coimbra.
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where 0 < q < 1 and the q-products are defined as