A few properties of the third Jackson q-Bessel function

On Basic Fourier-Bessel Expansions

“…Its prove is essentially coincident with the corresponding one of the [13] so we omit it. However, at least when µ = ν or µ = ν + 1, the above estimate for J µ qj kν ; q does not seem accurate.…”

“…where the last equality can be derived from [22,Proposition 3.5] or [13,Proposition 5]. The asymptotic behavior of the q-integral that appears in the Fourier coefficient (2.3), as well as the asymptotic behavior (as k → ∞) of the factors J ν qj kν ; q 2 and J ν j kν ; q 2 appearing in η k,ν , are crucial for further developments related with convergence issues of the Fourier-Bessel expansion (2.2).…”

“…Finally, using in (4.1) the hypothesis t we do not expect to prove a similar version for the case of the basic Fourier-Bessel expansions since we do not have, in this context, the nice properties and the formulary that the classical trigonometric functions satisfy, i.e., we do not expect to prove that lim With this regard see also [13,Remark 4,p. 13].…”

Uniform convergence of basic Fourier–Bessel series on a q-linear grid

“…For the properties of the more general spaces and , with , see [ 27 ]. We will also need the following formula of q -integration by parts [ 25 , Lemma 2, p. 327], valid for , assuming the involved limits exist: The third Jackson q -Bessel function has a countable infinite number of real and simple zeros [ 42 ]. In [ 4 , Theorem 2.3] it was proved that, when , the positive zeros of the function satisfy with where Using Taylor expansion it is plain that, as , The restriction can be dropped if k is chosen large enough [ 4 , Remark 2.5, p. 4247] because ( 2.5 )–( 2.7 ) remain valid for every whenever .…”

“…The Fourier–Bessel series on a q -linear grid associated with f is defined as the sum or, equivalently, with the coefficients given as and The last equality in formula ( 4.3 ) follows from the identity (see, e.g., [ 25 , Prop. 5 (vii), p. 330]) Theorem B assures mean convergence of the series ( 4.1 ).…”