Inequalities and bounds for the $ p $-generalized trigonometric functions

Li Yin; Liguo Huang; Xiuli Lin; Li Yin; Liguo Huang; Xiuli Lin

doi:10.3934/math.2021644

AIMS Mathematics

2021, Volume 6, Issue 10: 11097-11108. doi: 10.3934/math.2021644

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Research article

Inequalities and bounds for the $p$ -generalized trigonometric functions

1.
College of Science, Binzhou University, Binzhou, Shandong, 256603, China
2.
College of Mathematics Science, Qufu Normal University, Qufu, Shandong, 273165, China

Received: 24 April 2021 Accepted: 29 July 2021 Published: 03 August 2021
MSC : 33B10

In this paper, we mainly show some bounds and inequalities for the $p$ -generalized trigonometric functions defined by Richter.

Keywords:

$p$ -generalized trigonometric functions,
Turán type inequalities,
inequality

Citation: Li Yin, Liguo Huang, Xiuli Lin. Inequalities and bounds for the $p$ -generalized trigonometric functions[J]. AIMS Mathematics, 2021, 6(10): 11097-11108. doi: 10.3934/math.2021644

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Abstract

In this paper, we mainly show some bounds and inequalities for the $p$ -generalized trigonometric functions defined by Richter.

1. Introduction

Some special polynomials and numbers are diversely used in physics and engineering as well as in mathematics. For example, Bell polynomials play an important role in the studies of water waves which help energy development, mechanical engineering, marine/offshore engineering, hydraulic engineering, etc (see ^{[9,10,11,12,22]}). There are various ways of studying special numbers and polynomials, to mention a few, generating functions, combinatorial methods, $p$ -adic analysis, umbral calculus, differential equations, probability theory, special functions and analytic number theory (see ^[1,2]).

The aim of this paper is to introduce several special polynomials and numbers, and to study their explicit expressions, recurrence relations and identities involving those polynomials and numbers by using generating functions.

Indeed, we introduce Bell polynomials and numbers of the second kind (see (2.3), (2.5)) and poly-Bell polynomials and numbers of the second kind (see (4.1)). The generating function of Bell numbers of the second kind is the compositional inverse of the generating function of Bell numbers minus the constant term. Then Bell polynomials of the second kind are natural extensions of those numbers (see ^[23]). The poly-Bell polynomials of the second kind, which are defined with the help of polylogarithm, become the Bell polynomials of the second kind up to sign when the index of the polylogarithm is $k = 1$ .

We also consider degenerate versions of those numbers and polynomials, namely degenerate Bell numbers and polynomials of the second (see (3.3), (3.5)) and degenerate poly-Bell numbers and polynomials (see (5.1)), and derive similar results. It is worthwhile to note that degenerate versions of many special numbers and polynomials have been explored in recent years with aforementioned tools and many interesting arithmetical and combinatorial results have been obtained (see ^{[14,15,18,19,26]}). In fact, studying degenerate versions can be done not only for polynomials and numbers but also for transcendental functions like gamma functions. For the rest of this section, we recall the necessary facts that are needed throughout this paper.

The Stirling numbers of the first kind, $S_{1}(n, k)$ , are given by

$\begin{equation} \frac{1}{k!}\big(\log(1+t)\big)^{k} = \sum\limits_{n = k}^{\infty}S_{1}(n, k)\frac{t^{n}}{n!}, \quad (k\ge 0), \quad (\mathrm{see}\ [7, 25]), \end{equation}$

(1.1)

As the inversion formula of (1.1), the Stirling numbers of the second kind, $S_{2}(n, k),$ are given by

$\begin{equation} \frac{1}{k!}\big(e^{t}-1\big)^{k} = \sum\limits_{n = k}^{\infty}S_{2}(n, k)\frac{t^{n}}{n!}, \quad (k\ge 0), \quad (\mathrm{see}\ [3, 13-20]). \end{equation}$

(1.2)

It is well known that the Bell polynomials are defined as

$\begin{equation} \mathrm{Bel}_{n}(x) = \sum\limits_{k = 0}^{n}S_{2}(n, k)x^{k}, \quad (n\ge 0), \quad (\mathrm{see}\ [24, 25]). \end{equation}$

(1.3)

From (1.3), we note that

$\begin{equation} e^{x(e^{t}-1)} = \sum\limits_{n = 0}^{\infty}\mathrm{Bel}_{n}(x)\frac{t^{n}}{n!}, \quad (\mathrm{see}\ [4, 7, 8, 17, 27]). \end{equation}$

(1.4)

When $x = 1$ , $\mathrm{Bel}_{n} = \mathrm{Bel}_{n}(1), \ (n\ge 0)$ are called the Bell numbers.

For any $\lambda\in\mathbb{R}$ , the degenerate exponential function is given by

$\begin{equation} e_{\lambda}^{x}(t) = \sum\limits_{n = 0}^{\infty}\frac{(x)_{n, \lambda}}{n!}t^{n} , \quad (\mathrm{see}\ [5, 6, 7, 26, 27]), \end{equation}$

(1.5)

where $(x)_{0, \lambda} = 1, \ (x)_{n, \lambda} = x(x-\lambda)\cdots(x-(n-1)\lambda), \ (n\ge 1)$ .

When $x = 1$ , we write $e_{\lambda}(t) = e_{\lambda}^{1}(t)$ .

The degenerate Stirling numbers of the first kind are defined by

$\begin{equation} \frac{1}{k!}\big(\log_{\lambda}(1+t)\big)^{k} = \sum\limits_{n = k}^{\infty}S_{1, \lambda}(n, k)\frac{t^{n}}{n!}, \ (k\ge 0), \ (\mathrm{see}\ [15]), \end{equation}$

(1.6)

where

$\begin{equation} \log_{\lambda}(1+t) = \sum\limits_{n = 1}^{\infty}\lambda^{n-1}(1)_{n, 1/\lambda}\frac{t^{n}}{n!}, \quad (\mathrm{see}\ [15]). \end{equation}$

(1.7)

In view of (1.2), the degenerate Stirling numbers of the second kind are defined by

$\begin{equation} \frac{1}{k!}\big(e_{\lambda}(t)-1\big)^{k} = \sum\limits_{n = k}^{\infty}S_{2}(n, k)\frac{t^{n}}{n!}, \quad (\mathrm{see}\ [15]). \end{equation}$

(1.8)

In ^[17], the degenerate Bell polynomials are defined by

$\begin{equation} e_{\lambda}^{x}\big(e_{\lambda}(t)-1\big) = \sum\limits_{n = 0}^{\infty}\mathrm{Bel}_{n, \lambda}(x)\frac{t^{n}}{n!}. \end{equation}$

(1.9)

When $x = 1$ , $\mathrm{Bel}_{n, \lambda} = \mathrm{Bel}_{n, \lambda}(1), \ (n\ge 0)$ , are called the Bell numbers.

From (1.8) and (1.9), we note that

$\begin{equation} \mathrm{Bel}_{n, \lambda}(x) = \sum\limits_{k = 0}^{n}S_{2, \lambda}(n, k)(x)_{k, \lambda}, \ (n\ge 0), \ (\mathrm{see}\ [17]). \end{equation}$

(1.10)

The polylogarithm of index $k$ is given by

$\begin{equation} \mathrm{Li}_{k}(x) = \sum\limits_{n = 1}^{\infty}\frac{x^{n}}{n^{k}}, \quad (k\in\mathbb{Z}, \ |x| < 1), \quad(\mathrm{see}\ [3, 13, 14, 16, 21]). \end{equation}$

(1.11)

Note that $\mathrm{Li}_{1}(x) = -\log(1-x)$ .

Recently, the degenerate polylogarithm is defined as

$\begin{equation} \mathrm{Li}_{k, \lambda}(x) = \sum\limits_{n = 1}^{\infty}\frac{(-\lambda)^{n-1}(1)_{n, 1/\lambda}}{(n-1)!n^{k}}x^{n}, \quad(|x| < 1, \ k\in\mathbb{Z}), \quad(\mathrm{see}\ [15]). \end{equation}$

(1.12)

Note that $\mathrm{Li}_{1, \lambda}(x) = -\log_{\lambda}(1-x)$ .

Here we mention that, to our best knowledge, the results of this paper are new.

2. Bell polynomials of the second kind

From (1.4), we note that

$e^{x(e^{t}-1)} = \sum\limits_{n = 0}^{\infty}\mathrm{Bel}_{n}(x)\frac{t^{n}}{n!}$

Let $x = 1$ . Then we have

$\begin{equation} e^{e^{t}-1}-1 = \sum\limits_{n = 1}^{\infty}\mathrm{Bel}_{n}\frac{t^{n}}{n!}. \end{equation}$

(2.1)

Let $f(t) = e^{e^{t}-1}-1$ . Then the compositional inverse of $f(t)$ is given by

$\begin{equation} f^{-1}(t) = \log\big(1+\log(1+t)\big). \end{equation}$

(2.2)

We consider the new type Bell numbers, called Bell numbers of the second kind, defined by

$\begin{equation} \log\big(1+\log(1+t)\big) = \sum\limits_{n = 1}^{\infty}\mathrm{bel}_{n}\frac{t^{n}}{n!}. \end{equation}$

(2.3)

Now, we observe that

$\begin{align} &\log\big(1+\log(1+t)\big) = \sum\limits_{k = 1}^{\infty}\frac{(-1)^{k-1}}{k}\big(\log(1+t)\big)^{k} \\ & = \sum\limits_{k = 1}^{\infty}(-1)^{k-1}(k-1)!\frac{1}{k!}\big(\log(1+t)\big)^{k} = \sum\limits_{k = 1}^{\infty}(-1)^{k-1}(k-1)!\sum\limits_{n = k}^{\infty}S_{1}(n, k)\frac{t^{n}}{n!} \\ & = \sum\limits_{n = 1}^{\infty}\bigg(\sum\limits_{k = 1}^{n}(-1)^{k-1}(k-1)!S_{1}(n, k) \bigg)\frac{t^{n}}{n!}. \end{align}$

(2.4)

Therefore, by (2.3) and (2.4), we obtain the following theorem.

Theorem 1. For $n\ge 1$ , we have

$(-1)^{n-1}\mathrm{bel}_{n} = \sum\limits_{k = 1}^{n}(k-1)!{n \brack k},$

where ${n \brack k}$ are the unsigned Stirling numbers of the first kind.

Also, we consider the new type Bell polynomials, called Bell polynomials of the second kind, defined by

$\begin{equation} \mathrm{bel}_{n}(x) = \sum\limits_{k = 1}^{n}(-1)^{k-1}(k-1)!S_{1}(n, k)x^{k}, \quad (n\ge 1). \end{equation}$

(2.5)

From (2.5), we can derive the following equation.

$\begin{align} \sum\limits_{n = 1}^{\infty} \mathrm{bel}_{n}(x)\frac{t^{n}}{n!}& = \sum\limits_{n = 1}^{\infty}\bigg(\sum\limits_{k = 1}^{n}(-1)^{k-1}(k-1)!S_{1}(n, k)x^{k}\bigg)\frac{t^{n}}{n!} \\ & = \sum\limits_{k = 1}^{\infty}(-1)^{k-1}(k-1)!x^{k}\sum\limits_{n = k}^{\infty}S_{1}(n, k)\frac{t^{n}}{n!} \\ & = \sum\limits_{k = 1}^{\infty}\frac{(-1)^{k-1}k!}{k}x^{k}\frac{1}{k!}\big(\log(1+t)\big)^{k} \\ & = \sum\limits_{k = 1}^{\infty}\frac{(-1)^{k-1}}{k}x^{k}\big(\log(1+t)\big)^{k} = \log (1+x\log(1+t)). \end{align}$

(2.6)

Thus the generating function of Bell polynomials of the second kind is given by

$\begin{equation} \log\big(1+x\log(1+t)\big) = \sum\limits_{n = 1}^{\infty}\mathrm{bel}_{n}(x)\frac{t^{n}}{n!}. \end{equation}$

(2.7)

Note here that $\mathrm{bel}_{n} = \mathrm{bel}_{n}(1)$ . From (2.7), we note that

$\begin{equation} \frac{x}{\big(1+x\log(1+t)\big)(1+t)} = \frac{d}{dt}\log\big(1+x\log(1+t)\big) = \sum\limits_{n = 0}^{\infty}\mathrm{bel}_{n+1}(x)\frac{t^{n}}{n!} . \end{equation}$

(2.8)

Replacing $t$ by $e^{t}-1$ in (2.8), we get

$\begin{align} \frac{x}{1+xt}e^{-t}\ & = \ \sum\limits_{k = 0}^{\infty}\mathrm{bel}_{k+1}(x)\frac{1}{k!}\big(e^{t}-1\big)^{k} \\ & = \ \sum\limits_{k = 0}^{\infty}\mathrm{bel}_{k+1}(x)\sum\limits_{n = k} ^{\infty}S_{2}(n, k)\frac{t^{n}}{n!} \\ & = \ \sum\limits_{n = 0}^{\infty}\bigg(\sum\limits_{k = 0}^{n}\mathrm{bel}_{k+1}(x)S_{2}(n, k)\bigg)\frac{t^{n}}{n!}. \end{align}$

(2.9)

Taking $x = -1$ in (2.9), we have

$\begin{equation} \sum\limits_{n = 0}^{\infty}\bigg(\sum\limits_{k = 0}^{n}\mathrm{bel}_{k+1}(-1)S_{2}(n, k)\bigg)\frac{t^{n}}{n!} = -\frac{1}{1-t}e^{-t} = -\sum\limits_{n = 0}^{\infty}d_{n}\frac{t^{n}}{n!}, \end{equation}$

(2.10)

where $d_{n}$ is the derangement number (see ^[19]).

Therefore, by comparing the coefficients on both sides of (2.10), we obtain the following theorem.

Theorem 2. For $n\ge 0$ , we have

$\sum\limits_{k = 0}^{n}\mathrm{bel}_{k+1}(-1)S_{2}(n, k) = -d_{n}.$

Replacing $t$ by $e^{e^{t}-1}-1$ in (2.3), we get

$\begin{align} t& = \sum\limits_{k = 1}^{\infty}\mathrm{bel}_{k}\frac{1}{k!}\Big(e^{e^{t}-1}-1\Big)^{k} = \sum\limits_{k = 1}^{\infty}\mathrm{bel}_{k}\sum\limits_{j = k}^{\infty}S_{2}(j, k)\frac{1}{j!} \big(e^{t}-1\big)^{j} \\ & = \sum\limits_{j = 1}^{\infty}\sum\limits_{k = 1}^{j}\mathrm{bel}_{k}S_{2}(j, k)\sum\limits_{n = j}^{\infty}S_{2}(n, k)\frac{t^{n}}{n!} \\ & = \sum\limits_{n = 1}^{\infty}\bigg(\sum\limits_{j = 1}^{n}\sum\limits_{k = 1}^{j}\mathrm{bel}_{k}S_{2}(j, k)S_{2}(n, j)\bigg)\frac{t^{n}}{n!}. \end{align}$

(2.11)

Thus we obtain following theorem.

Theorem 3. For $n\ge 2$ , we have

$\sum\limits_{j = 1}^{n}\sum\limits_{k = 1}^{j}\mathrm{bel}_{k}S_{2}(j, k)S_{2}(n, j) = 0, \, \, \, \mathrm{and}\, \, \, \mathrm{bel}_{1} = 1.$

Replacing $t$ by $e^{t}-1$ in (2.7), we get

$\begin{align} \log(1+xt)\ & = \ \sum\limits_{k = 1}^{\infty}\mathrm{bel}_{k}(x)\frac{1}{k!}\big(e^{t}-1\big)^{k} \\ & = \ \sum\limits_{k = 1}^{\infty}\mathrm{bel}_{k}(x)\sum\limits_{n = k}^{\infty}S_{2}(n, k)\frac{t^{n}}{n!} \\ & = \ \sum\limits_{n = 1}^{\infty}\bigg(\sum\limits_{k = 1}^{n}\mathrm{bel}_{k}(x)S_{2}(n, k)\bigg)\frac{t^{n}}{n!}. \end{align}$

(2.12)

On the other hand,

$\begin{equation} \log(1+xt) = \sum\limits_{n = 1}^{\infty}\frac{(-1)^{n-1}}{n}x^{n}t^{n}. \end{equation}$

(2.13)

Therefore, by (2.12) and (2.13), we obtain the following theorem.

Theorem 4. For $n\ge 1$ , we have

$x^{n} = \frac{(-1)^{n-1}}{(n-1)!}\sum\limits_{k = 1}^{n}\mathrm{bel}_{k}(x)S_{2}(n, k).$

In particular,

$1 = \frac{(-1)^{n-1}}{(n-1)!}\sum\limits_{k = 1}^{n}\mathrm{bel}_{k}S_{2}(n, k).$

3. Degenerate Bell polynomials of the second kind

From (1.3), we note that

$\begin{equation} e_{\lambda}\big(e_{\lambda}(t)-1\big)-1 = \sum\limits_{n = 1}^{\infty}\mathrm{Bel}_{n, \lambda}\frac{t^{n}}{n!}. \end{equation}$

(3.1)

Let $f_{\lambda}(t) = e_{\lambda}\big(e_{\lambda}(t)-1\big)-1$ . Then the compositional inverse of $f_{\lambda}(t)$ is given by

$\begin{equation} f_{\lambda}^{-1}(t) = \log_{\lambda}\big(1+\log_{\lambda}(1+t)\big). \end{equation}$

(3.2)

We consider the new type degenerate Bell numbers, called degenerate Bell numbers of the second kind, defined by

$\begin{equation} \log_{\lambda}\big(1+\log_{\lambda}(1+t)\big) = \sum\limits_{n = 1}^{\infty}\mathrm{bel}_{n, \lambda}\frac{t^{n}}{n!}. \end{equation}$

(3.3)

Now, we observe that

$\begin{align} \log_{\lambda}\big(1+\log_{\lambda}(1+t)\big)\ & = \ \sum\limits_{k = 1}^{\infty}\lambda^{k-1}(1)_{k, 1/\lambda}\frac{1}{k!}\big(\log_{\lambda}(1+t)\big)^{k} \\ & = \ \sum\limits_{k = 1}^{\infty}\lambda^{k-1}(1)_{k, 1/\lambda}\sum\limits_{n = k}^{\infty}S_{1, \lambda}(n, k)\frac{t^{n}}{n!}. \\ & = \ \sum\limits_{n = 1}^{\infty}\bigg(\sum\limits_{k = 1}^{n}\lambda^{k-1}(1)_{k, 1/\lambda}S_{1, \lambda}(n, k)\bigg)\frac{t^{n}}{n!}. \end{align}$

(3.4)

Therefore, by (3.3) and (3.4), we obtain the following theorem.

Theorem 5. For $n\ge 1$ , we have

$\mathrm{bel}_{n, \lambda} = \sum\limits_{k = 1}^{n}\lambda^{k-1}(1)_{k, 1/\lambda}S_{1, \lambda}(n, k).$

Also, we define the degenerate Bell polynomials of second kind by

$\begin{equation} \mathrm{bel}_{n, \lambda}(x) = \sum\limits_{k = 1}^{n}\lambda^{k-1}(1)_{k, 1/\lambda}S_{1, \lambda}(n, k)x^{k}. \end{equation}$

(3.5)

Note that $\mathrm{bel}_{n, \lambda} = \mathrm{bel}_{n, \lambda}(1)$ .

From (3.5), we note that

$\begin{align} \sum\limits_{n = 1}^{\infty}\mathrm{bel}_{n, \lambda}(x)\frac{t^{n}}{n!}\ & = \ \sum\limits_{n = 1}^{\infty}\bigg(\sum\limits_{k = 1}^{n}\lambda^{k-1}(1)_{k, 1/\lambda}S_{1, \lambda}(n, k)x^{k}\bigg)\frac{t^{n}}{n!} \\ & = \ \sum\limits_{k = 1}^{\infty}\lambda^{k-1}(1)_{k, 1/\lambda}x^{k}\sum\limits_{n = k}^{\infty}S_{1, \lambda}(n, k)\frac{t^{n}}{n!} \\ & = \ \sum\limits_{k = 1}^{\infty}\lambda^{k-1}(1)_{k, 1/\lambda}x^{k}\frac{1}{k!}\big(\log_{\lambda}(1+t)\big)^{k} \\ & = \ \log_{\lambda}\big(1+x\log_{\lambda}(1+t)\big). \end{align}$

(3.6)

Thus the generating function of $\mathrm{bel}_{n, \lambda}(x)$ is given by

$\begin{equation} \log_{\lambda}\big(1+x\log_{\lambda}(1+t)\big) = \sum\limits_{n = 1}^{\infty}\mathrm{bel}_{n, \lambda}(x)\frac{t^{n}}{n!}. \end{equation}$

(3.7)

Replacing $t$ by $e_{\lambda}(t)-1$ in (3.7), we get

$\begin{align} \log_{\lambda}(1+xt)\ & = \ \sum\limits_{k = 1}^{\infty}\mathrm{bel}_{k, \lambda}(x)\frac{1}{k!}\big(e_{\lambda}(t)-1\big)^{k} \\ & = \ \sum\limits_{k = 1}^{\infty}\mathrm{bel}_{k, \lambda}(x)\sum\limits_{n = k}^{\infty}S_{2, \lambda}(n, k)\frac{t^{n}}{n!} \\ & = \ \sum\limits_{n = 1}^{\infty}\bigg(\sum\limits_{k = 1}^{n}\mathrm{bel}_{k, \lambda}(x)S_{2, \lambda}(n, k)\bigg)\frac{t^{n}}{n!}. \end{align}$

(3.8)

On the other hand,

$\begin{equation} \log_{\lambda}(1+xt) = \sum\limits_{n = 1}^{\infty}\lambda^{n-1}(1)_{n, 1/\lambda}x^{n}\frac{t^{n}}{n!}. \end{equation}$

(3.9)

Therefore, by (3.8) and (3.9), we obtain the following theorem.

Theorem 6. For $n\ge 1$ , we have

$x^{n} = \frac{\lambda^{1-n}}{(1)_{n, 1/\lambda}}\sum\limits_{k = 1}^{n}\mathrm{bel}_{k, \lambda}(x)S_{2, \lambda}(n, k).$

In particular,

$\lambda^{n-1}(1)_{n, 1/\lambda} = \sum\limits_{k = 1}^{n}\mathrm{bel}_{k, \lambda}S_{2, \lambda}(n, k).$

Replacing $t$ by $e_{\lambda}\big(e_{\lambda}(t)-1\big)-1$ in (3.3), we have

$\begin{align} t\ & = \sum\limits_{k = 1}^{\infty}\mathrm{bel}_{k, \lambda}\frac{1}{k!} \big(e_{\lambda}(e_{\lambda}(t)-1)-1\big)^{k} \\ & = \sum\limits_{k = 1}^{\infty}\mathrm{bel}_{k, \lambda}\sum\limits_{j = k}^{\infty}S_{2, \lambda}(j, k)\frac{1}{j!}\big(e_{\lambda}(t)-1\big)^{j} \\ & = \ \sum\limits_{j = 1}^{\infty}\bigg(\sum\limits_{k = 1}^{j}\mathrm{bel}_{k, \lambda}S_{2, \lambda}(j, k)\bigg)\sum\limits_{n = j}^{\infty}S_{2, \lambda}(n, j)\frac{t^{n}}{n!} \\ & = \ \sum\limits_{n = 1}^{\infty}\bigg(\sum\limits_{j = 1}^{n}\sum\limits_{k = 1}^{j}\mathrm{bel}_{k, \lambda}S_{2, \lambda}(j, k)S_{2, \lambda}(n, j)\bigg)\frac{t^{n}}{n!}. \end{align}$

(3.10)

Therefore, by comparing the coefficients on both sides of (3.10), we obtain the following theorem.

Theorem 7. For $n\ge 2$ , we have

$\sum\limits_{j = 1}^{n}\sum\limits_{k = 1}^{j}\mathrm{bel}_{k, \lambda}S_{2, \lambda}(j, k)S_{2, \lambda}(n, j) = 0, \, \, \, \mathrm{and}\, \, \, \mathrm{bel}_{1, \lambda} = 1.$

4. Poly-Bell polynomials of the second kind

Now, we consider the poly-Bell polynomials of the second kind which are defined as

$\begin{equation} \mathrm{Li}_{k}\big(-x\log(1-t)\big) = \sum\limits_{n = 1}^{\infty}\mathrm{bel}_{n}^{(k)}(x)\frac{t^{n}}{n!}. \end{equation}$

(4.1)

When $x = 1$ , $\mathrm{bel}_{n}^{(k)} = \mathrm{bel}_{n}^{(k)}(1)$ are called the poly-Bell numbers of the second kind.

From (1.11), we note that

$\begin{align} \mathrm{Li}_{k}\big(-x\log(1-t)\big)\ & = \ \sum\limits_{l = 1}^{\infty}\frac{(-1)^{l}}{l^{k}}x^{l}l!\frac{1}{l!}\big(\log(1-t)\big)^{l} \\ & = \ \sum\limits_{l = 1}^{\infty}\frac{(-1)^{l}}{l^{k-1}}(l-1)!x^{l}\sum\limits_{n = l}^{\infty}(-1)^{n}S_{1}(n, l)\frac{t^{n}}{n!} \\ & = \ \sum\limits_{n = 1}^{\infty}\bigg(\sum\limits_{l = 1}^{n}\frac{(-1)^{n-l}}{l^{k-1}}(l-1)!x^{l}S_{1}(n, l)\bigg)\frac{t^{n}}{n!}. \end{align}$

(4.2)

Therefore, by (4.1) and (4.2), we obtain the following theorem.

Theorem 8. For $n\ge 1$ , we have

$\mathrm{bel}_{n}^{(k)}(x) = \sum\limits_{l = 1}^{n}\frac{x^{l}}{l^{k-1}}(l-1)!{n \brack l}.$

In particular,

$\mathrm{bel}_{n}^{(k)} = \sum\limits_{l = 1}^{n}\frac{1}{l^{k-1}}(l-1)!{n \brack l}.$

Note that

$\mathrm{bel}_{n}^{(1)}(x) = \sum\limits_{l = 1}^{n}x^{l}(l-1)! {n \brack l} = (-1)^{n-1}\mathrm{bel}_{n}(x).$

Indeed,

$\begin{align*} \mathrm{Li}_{1}\big(-x\log(1-t)\big)\ & = \ -\log\big(1+x\log(1-t)\big) \\ & = \ \sum\limits_{n = 1}^{\infty}\mathrm{bel}_{n}(x)(-1)^{n-1}\frac{t^{n}}{n!}. \end{align*}$

Replacing $t$ by $1-e^{-t}$ in (4.1), we get

$\begin{align} \mathrm{Li}_{k}(xt)\ & = \ \sum\limits_{l = 1}^{\infty}\mathrm{bel}_{l}^{(k)}(x)\frac{1}{l!}\big(1-e^{-t}\big)^{l} \\ & = \ \sum\limits_{l = 1}^{\infty}\mathrm{bel}_{l}^{(k)}(x)(-1)^{l}\sum\limits_{n = l}^{\infty}S_{2}(n, l)(-1)^{n}\frac{t^{n}}{n!}. \\ & = \ \sum\limits_{n = 1}^{\infty}\bigg(\sum\limits_{l = 1}^{n}(-1)^{n-l}\mathrm{bel}_{l}^{(k)}(x)S_{2}(n, l)\bigg)\frac{t^{n}}{n!}. \end{align}$

(4.3)

From (1.11) and (4.3), we note that

$\begin{equation} \frac{x^{n}}{n^{k}} = \frac{1}{n!}\sum\limits_{l = 1}^{n}(-1)^{n-l}\mathrm{bel}_{l}^{(k)}(x)S_{2}(n, l). \end{equation}$

(4.4)

Therefore, by (4.4), we obtain the following theorem.

Theorem 9. For $n\ge 1$ , we have

$x^{n} = \frac{n^{k-1}}{(n-1)!}\sum\limits_{l = 1}^{n}(-1)^{n-l}\mathrm{bel}_{l}^{(k)}(x)S_{2}(n, l).$

5. Degenerate poly-Bell polynomials of the second kind

We define the degenerate poly-Bell polynomials of the second kind by

$\begin{equation} \mathrm{Li}_{k, \lambda}\big(-x\log_{\lambda}(1-t)\big) = \sum\limits_{n = 1}^{\infty}\mathrm{bel}_{n, \lambda}^{(k)}(x)\frac{t^{n}}{n!}. \end{equation}$

(5.1)

When $x = 1$ , $\mathrm{bel}_{n, \lambda}^{(k)} = \mathrm{bel}_{n, \lambda}^{(k)}(1)$ are called the degenerate poly-Bell numbers of the second.

From (2.1), we note that

$\begin{align} \mathrm{Li}_{k, \lambda}\big(-x\log_{\lambda}(1-t)\ & = \ \sum\limits_{l = 1}^{\infty}\frac{(-\lambda)^{l-1}(1)_{l, 1/\lambda}}{(l-1)!l^{k}}\big(-x\log_{\lambda}(1-t)\big)^{l} \\ & = \ -\sum\limits_{l = 1}^{\infty}\frac{(1)_{l, 1/\lambda}}{l^{k-1}}\lambda^{l-1}x^{l}\frac{1}{l!}\big(\log_{\lambda}(1-t)\big)^{l}. \\ & = -\sum\limits_{l = 1}^{\infty}\frac{(1)_{l, 1/\lambda}}{l^{k-1}}\lambda^{l-1}x^{l}\sum\limits_{n = l}^{\infty}S_{1, \lambda}(n, l)(-1)^{n}\frac{t^{n}}{n!} \\ & = \ \sum\limits_{n = 1}^{\infty}\bigg((-1)^{n-1}\sum\limits_{l = 1}^{n}\frac{1}{l^{k-1}}(1)_{l, 1/\lambda}\lambda^{l-1}x^{l}S_{1, \lambda}(n, l)\bigg)\frac{t^{n}}{n!}. \end{align}$

(5.2)

Therefore, by (5.1) and (5.2), we obtain the following theorem.

Theorem 10. For $n\ge 1$ , we have

$(-1)^{n-1}\mathrm{bel}_{n, \lambda}^{(k)}(x) = \sum\limits_{l = 1}^{n}\frac{1}{l^{k-1}}(1)_{l, 1/\lambda}\lambda^{l-1}x^{l}S_{1, \lambda}(n, l).$

For $k = 1$ , we have

$(-1)^{n-1}\mathrm{bel}_{n, \lambda}^{(1)}(x) = \sum\limits_{l = 1}^{n}(1)_{l, 1/\lambda}\lambda^{l-1}x^{l}S_{1, \lambda}(n, l) = \mathrm{bel}_{n, \lambda}(x), \quad (n\ge 0).$

Indeed,

$\mathrm{Li}_{1, \lambda}\big(-x\log_{\lambda}(1-t)\big) = -\log_{\lambda}\big(1+x\log_{\lambda}(1-t)\big) = \sum\limits_{n = 1}^{\infty}(-1)^{n-1}\mathrm{bel}_{n, \lambda}(x)\frac{t^{n} }{n!}.$

Replacing $t$ by $1-e_{\lambda}(-t)$ in (5.1), we get

$\begin{align} \mathrm{Li}_{k, \lambda}(xt)\ & = \ \sum\limits_{l = 1}^{\infty} \mathrm{bel}_{l, \lambda}^{(k)}(x)\frac{1}{l!}\big(1-e_{\lambda}(-t)\big)^{l} \\ & = \ \sum\limits_{l = 1}^{\infty} \mathrm{bel}_{l, \lambda}^{(k)}(x)(-1)^{l}\frac{1}{l!}\big(e_{\lambda}(-t)-1\big)^{l} \\ & = \ \sum\limits_{l = 1}^{\infty} \mathrm{bel}_{l, \lambda}^{(k)}(x)(-1)^{l}\sum\limits_{n = l}^{\infty}S_{2, \lambda}(n, l)(-1)^{n}\frac{t^{n}}{n!} \\ & = \ \sum\limits_{n = 1}^{\infty}\bigg(\sum\limits_{l = 1}^{n}(-1)^{n-l} \mathrm{bel}_{l, \lambda}^{(k)}(x)S_{2, \lambda}(n, l)\bigg)\frac{t^{n}}{n!}. \end{align}$

(5.3)

On the other hand,

$\begin{equation} \mathrm{Li}_{k, \lambda}(xt) = \sum\limits_{n = 1}^{\infty}\frac{(-\lambda)^{n-1}(1)_{n, 1/\lambda}}{(n-1)!n^{k}}x^{n}t^{n} = \sum\limits_{n = 1}^{\infty}\frac{(-\lambda)^{n-1}(1)_{n, 1/\lambda}}{n^{k-1}}x^{n}\frac{t^{n}}{n!}. \end{equation}$

(5.4)

From (5.3) and (5.4), we get the following result.

Theorem 11. For $n \ge 1$ , we have

$\frac{(-\lambda)^{n-1}(1)_{n, 1/\lambda}}{n^{k-1}}x^{n} = \sum\limits_{l = 1}^{n}(-1)^{n-l}\mathrm{bel}_{l, \lambda}^{(k)}(x)S_{2, \lambda}(n, l).$

6. Conclusions

Many special polynomials and numbers are widely used in physics and engineering as well as in mathematics. In recent years, degenerate versions of some special polynomials and numbers have been studied by means of various different tools. Here we introduced Bell polynomials of the second kind, poly-Bell polynomials of the second kind and their degenerate versions, namely degenerate Bell polynomials of the second kind and degenerate poly-Bell polynomials of the second kind. By using generating functions, we explored their explicit expressions, recurrence relations and some identities involving those polynomials and numbers.

It is one of our future projects to continue this line of research, namely to explore many special numbers and polynomials and their degenerate versions with the help of various different tools.

Acknowledgments

This work was supported by the Basic Science Research Program, the National Research Foundation of Korea, (NRF-2021R1F1A1050151).

Conflict of interest

The authors declare no conflict of interest.

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AIMS Mathematics

Inequalities and bounds for the $p$ -generalized trigonometric functions

Related Papers:

Abstract

1. Introduction

2. Bell polynomials of the second kind

3. Degenerate Bell polynomials of the second kind

4. Poly-Bell polynomials of the second kind

5. Degenerate poly-Bell polynomials of the second kind

6. Conclusions

Acknowledgments

Conflict of interest

References

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AIMS Mathematics

Inequalities and bounds for the p p -generalized trigonometric functions

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Abstract

1. Introduction

2. Bell polynomials of the second kind

3. Degenerate Bell polynomials of the second kind

4. Poly-Bell polynomials of the second kind

5. Degenerate poly-Bell polynomials of the second kind

6. Conclusions

Acknowledgments

Conflict of interest

References

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Inequalities and bounds for the $p$ -generalized trigonometric functions