Some explicit identities for the modified higher-order degenerate q-Euler polynomials and their zeroes

Abstract

In this paper, we define the modified higher-order degenerate q-Euler polynomials and give some identities for these polynomials. Also we give numerical investigations of the zeroes of the modified higher-order q-Euler polynomials and the zeroes of the modified higher-order degenerate q-Euler polynomials.Furthermore, we demonstrate the shapes and zeroes of the modified higher-order q-Euler polynomials and the modified higher-order degenerate q-Euler polynomials by using a computer.

Sheffer type degenerate Euler and Bernoulli polynomials

“…In Figure 1 (second from left), we choose n = 30, λ = 1 10 , and q = 3. In Figure 1 (second from right), we choose n = 30, λ = 5 10 , and q = 1 3 . In Figure 1 (first from right), we choose n = 30, λ = 5 10 , and q = 3.…”

“…In Figure 1 (second from right), we choose n = 30, λ = 5 10 , and q = 1 3 . In Figure 1 (first from right), we choose n = 30, λ = 5 10 , and q = 3. We plot the zeros of the Sheffer type degenerate Euler polynomials ε (C) n,λ (p, q)( Figure 2).…”

“…where f (t) = n i=0 a i t i , (a 0 0) and (t) = ∞ i=0 a i t i , (a 1 0). It is well known that the most famous Sheffer polynomials are the Bernoulli polynomials and the Euler polynomials: the Bernoulli polynomials are defined by the generating function to be (see [1][2][3][4][5][6]) t e t − 1…”

“…. (5) Note that lim λ→∞ β n,λ (x) = B n (x) and lim λ→∞ ε n,λ (x) = E n (x). When x = 0, β n,λ = β n,λ (0) and ε n,λ = ε n,λ (0) are called degenerate Bernoulli and Euler numbers, respectively.…”

Representation by Degenerate Genocchi Polynomials

“…Many interesting identities have been derived by using similar formulas for representations by Bernoulli, Euler, and Frobenius-Euler polynomials (see [1][2][3][4][5][6][7][8][9]). e list in the references is far from being exhaustive.…”