λ-Analogues of Stirling polynomials of the first kind and their applications

Abstract

In this paper, we study λ-analogues of the r-Stirling numbers of the first kind which have close connections with the r-Stirling numbers of the first kind and λ-Stirling numbers of the first kind. Specifically, we give the recurrence relations for these numbers and show their connections with the λ-Stirling numbers of the first kind and higher-order Daehee polynomials.2010 Mathematics Subject Classification. 11B73; 11B83. Key words and phrases. λ-analogues of the r-Stirling numbers of the first kind, higher-order Daehee polynomials.

Degenerate polyexponential-Genocchi numbers and polynomials

“…n,λ (0) are called the degenerate poly-Bernoulli numbers. The Daehee polynomials [28] are defined by…”

“…standing for the Stirling numbers of the second kind given by means of the following generating function: [14,15,18,[20][21][22][23][24][25][26][27][28][29][30][31][32]).…”

Some Properties of the Degenerate Hyperbolic Functions

“…for λ ∈ (0, ∞) and t ∈ R. Figures 1,2,3,4,5 and 6 below, are the plots of the degenerate hyperbolic cosine, sine, tangent, secant, cosecant and cotangent functions respectively. The introduction of the degenerate hyperbolic functions was motivated by the degenerate exponential function, which is well known in the literature (see [2], [3], [4], [5], [6], [7]), and the recent interest of many reseachers in establishing degenerate versions of some special functions(see [8], [9]). In [10], the hyperbolic secant function has been applied to handle noise in data processing.…”

Normal ordering associated with λ-Stirling numbers inλ-Shift algebra

“…In [16], introduced are a λ -analogue of the unsigned Stirling numbers of the first kind n k λ and that of the unsigned r-Stirling numbers of the first kind n k r,λ respectively as a λ -analogue of n k and that of n k r , (see (8), (9)). The Stirling numbers of the second kind appear as the coefficients in normal orderings in the Weyl algebra (see (10), (11)), while the unsigned Stirling numbers of the first kind appear as those in normal orderings in the shift algebra S (see (12), (13)).…”

“…The Weyl algebra is the unital algebra generated by letters a and a † satisfying the commutation aa † − a † a = 1, (see [1,2,[7][8][9][10][11][12][13][14][15][16][17][18][19][20][21]). (10) Katriel proved that the normal ordering in Weyl algebra is given by (see ( 6))…”