A Class of Index Transforms with Whittaker's Function as the Kernel

The Cherry Transform and its Relationship with a Singular Sturm-liouville Problem

“…In this sequel to the recent paper [7], we first show that the Cherry expansion in (1.7) is, in fact, an eigenfunction expansion arising from a certain singular Sturm-Liouville boundary value problem associated with the differential equation (13). Our assertion then enables us to prove, in Section 2, a Plancherel-type theorem for the Cherry transform.…”

“…the Cherry transform in (1.7) involves integration with respect to the first index of the Whittaker function, and (therefore) differs from the index transform considered recently by Srivastava et al [7] (see also [10], [13] and [15, p. 203]). Other classes of integral transforms with Whittaker's function as their kernels include (for example) those of the convolution type and related to the Mellin convolution (cf.…”

The spectral expansion approach to index transforms and connections with the theory of diffusion processes

“…where the integral converges with respect to the norm in L 2 (0, ∞); dy y . The so-called index Whittaker transform is the integral transform [27,32]…”

“…Consider the generalized Yor integral for the index Whittaker transform with parameter α < 1 2 , defined by (3.34). In order to show that this generalized Yor integral obeys the evolution equation, we start by estimating ϑ(t, x): using the inequality |W α,iτ (2y)| ≤ C Γ( 1 2 − α + iτ ) −2 (2y) α e −y (see [27], Eq. (1.15)) we deduce that |ϑ(t, x)| ≤ Cπ −5/2 (2x) α− 1 2 e −x ∞ 0 e −t(τ 2 +α 2 ) τ sinh(2πτ )dτ.…”

On the product formula and convolution associated with the index Whittaker transform

“…Therefore, the fact that Ψ a is an isomorphism and the inversion formula follows from known results on the L 2 -theory for the index Whittaker transform, cf. [25,Section 3]. As for relation (44), it is due to the fact that the index Whittaker transform in confluent hypergeometric form arises from the spectral expansion of the differential operator (42), cf.…”