Asymptotics and Confluence for Some Linear q-Difference–Differential Cauchy Problem

Solutions of a Class of Multiplicatively Advanced Differential Equations II: Fourier Transforms

“…where (27) was used to move from (45) to ( 46) and (33) in Lemma 3 was used to move from (46) to (47). Thus, for k ≥ 0, the residue of…”

“…There is a robust study within the area of q-difference differential equations with dilations involving q > 1. This is highlighted by works of L. Di Vizio [10][11][12]; C. Hardouin [11]; T. Dreyfus [13,14]; A. Lastra [14], [15][16][17][18][19][20], [21][22][23]; S. Malek [14], [15][16][17][18][19][20], [21][22][23], [24][25][26][27]; J. Sanz [21][22][23]; H. Tahara [28]; and C. Zhang [12,29], along with further references by these researchers and others. Also, for good background references to the current work, consult [2][3][4][5][6][30][31][32][33][34] (especially [2,4]).…”

Multisummability of formal solutions for a family of generalized singularly perturbed moment differential equations

“…q = [1] q [2] q • • • [p] q is known as the sequence of q−factorials, where [ ] q = −1 j=0 q j . It determines the moment differentiation which coincides with the q−derivative D q,z given by D q,z z p = [p] q z p−1 for every p ∈ N. This moment differentiation is quite related to the dilation operator, appearing in the study of q−difference equations which is of great interest in the scientific community with interesting advances in the knowledge of the asymptotic behavior of the solutions of q−difference equations (see [9,15] among others, and the references therein).…”

Asymptotics and Confluence for a Singular Nonlinearq-Difference-Differential Cauchy Problem

“…The present work is the sequel of the investigation initiated in [1] that focused on some linear q − differencedifferential Cauchy problems outlined as…”

“…where S ≥ 1 is a suitable natural number, the piece PðTÞ from the leading term of (3) is the polynomial appearing in (1), k ≥ 1 is the integer stemming from (1), σ q;t stands for the dilation operator acting on functions through σ q;t f ðt, zÞ = f ðqt, zÞ for q > 1 arising in (1) and the right handside P ðt, z, V 1 , V 2 , V 3 Þ together with the data (4) represent fittingly selected polynomials. As summed up in Theorem 3 of this work, under strong restrictions on the shape of (3) (not discussed in this paper but listed in [1]), a finite set fu p ðt, zÞg 0≤p≤ς−1 , for some integer ς ≥ 2, of holomorphic solutions to (3) and ( 4) could be modeled on products T p × D, where D stands for some small disc centered at 0 in ℂ and where T = fT p g 0≤p≤ς−1 is a suitable set of bounded sectors whose union covers some neighborhood of 0 in ℂ * , see Definition 1. These solutions were expressed through Laplace transforms of order k,…”