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Identities and recurrence relations of special numbers and polynomials of higher order by analysis of their generating functions
Journal of Inequalities and Applications volume 2018, Article number: 220 (2018)
Abstract
The aim of this is to give generating functions for new families of special numbers and polynomials of higher order. By using these generating functions and their functional equations, we derive identities and relations for these numbers and polynomials. Relations between these new families of special numbers and polynomials and Bernoulli numbers and polynomials are given. Finally, recurrence relations and derivative formulas, which are related to these numbers and polynomials, are given.
1 Introduction
Many special numbers and polynomials have been found by researchers in recent years. These numbers and polynomials have various applications in mathematics and related areas. It is well known that special polynomials have been applied to model including real phenomena (cf. [1–15]; see also the references cited therein). In this work, we define a new family of special numbers and polynomials of higher order with their generating functions. Therefore, motivation of this paper is to derive some identities and recurrence relations for these new families of special numbers and polynomials of higher order by using generating functions, functional equations, and partial derivative formulas.
In order to derive the results of this paper, we need generating functions for new families of special numbers and polynomials.
Kim et al. [3, Eq. (2.5)] defined the following generating function for the special numbers \(\mathfrak{s}_{n} ( a;q ) \):
By help of the above generating function, Kim et al. [3, Eq. (2.5)] also defined the following generating function for the special polynomials \(\mathfrak{s}_{n} ( x;a;q ) \):
We summarize the content of this paper as follows:
In Sect. 2, relations between a new family of specials numbers and polynomials, Apostol–Bernoulli numbers and polynomials, and also Bernoulli numbers and polynomials are given.
In Sect. 3, generating functions for new families of special numbers and polynomials of higher order are given. By using these functions with their functional equations, many formulas, identities, and relations for these numbers and polynomials are given.
In Sect. 4, by using partial derivative formulas for generating functions of new families of special numbers and polynomials of higher order, recurrence relations for these numbers and polynomials are given. Finally, we complete this paper with conclusion section.
2 Relations between Bernoulli type numbers and polynomials
We give relations between Apostol–Bernoulli numbers and polynomials and also Bernoulli numbers and polynomials.
Apostol Bernoulli polynomials of order k are defined by the following generating function:
When \(x=0\), we have the well-known numbers which are called Apostol–Bernoulli numbers of order k. That is,
observe that
(cf. [1–15]; see also the references cited therein).
Substituting \(a=e\) into (1) and (2), relations between Apostol–Bernoulli numbers and polynomials are given as follows:
From the above equation, we get
Now, we equate the coefficients of \(\frac{t^{n}}{n!}\) on both sides of the above equation, we deduce the following relation.
Theorem 1
If \(n\geq1\) is a positive integer, then we obtain
When \(q\rightarrow1\), equation (2) reduces to a generating function for Bernoulli polynomials \(B_{n} (x)\). That is,
Therefore, we have
From the above equation, we have
It is now clear that Bernoulli numbers \(B_{n}\) are related to the numbers \(\mathfrak{s}_{n} ( x;a;q )\). That is,
3 New families of special numbers and polynomials of higher order
In this section, we give generating functions for new families of special numbers and polynomials of high order. By using these functions with their functional equations, we derive some formulas, identities, and relations for these numbers and polynomials.
Generating functions for new families of special numbers of order k are given by
Observe that when \(k=1\), we have
(cf. [3]).
We now give a computation formula for the numbers \(\frak{s}_{n}^{ ( k ) } ( a;q ) \) by the following theorem.
Theorem 2
Let k and v be nonnegative integers. Then we have
Proof
We set the following functional equation:
Combining the above equation with (4), we have
Therefore
Equating the coefficients of \(\frac{t^{n}}{n!}\) on both sides of the above equation, we arrive at the desired result. □
Observe that when \(k=v=1\), (6) reduces to the following formula:
Setting \(n=0\) in (6) the above equation, we get
Setting \(n=1\) in (6), we get
With the help of (4), we define the following generating function for a new family of polynomials:
By combining (8) with (4), we get
Therefore
Equating the coefficients of \(\frac{t^{n}}{n!}\) on both sides of the above equations, we get the following theorem.
Theorem 3
If n and k are nonnegative integers, then
By using (9), we easily see that
Setting \(k=0\) into (9), we get
Substituting \(x=u+w\) into (8), we get
and
By using the above equations, we derive the following formulas, respectively:
and
Combining (10) with (11), we get the following theorem.
Theorem 4
If n is a nonnegative integer, then
Substituting \(w=1\) into (12), we get
Thus we also get the following results:
and
From the above equations, we obtain the following corollary.
Corollary 5
If n is a nonnegative integer, then
4 Recurrence relations and derivative formulas
In this section, by using partial derivative formulas for generating functions of new families of special numbers and polynomials of higher order, we derive recurrence relations for these numbers and polynomials.
Theorem 6
If n is a nonnegative integer, then
Proof
Taking partial derivative of Equation (8) with respect to x, we get the following derivative formula:
By combining the above equation with (8), we obtain
Making some elementary calculations in the above equation after equating the coefficients of \(\frac{t^{n}}{n!}\) on both sides of the final equation, we arrive at the desired result. □
Theorem 7
(Recurrence relation)
If n is a nonnegative integer, then
Proof
Taking partial derivative of Equation (4) with respect to t, we get the following derivative formula:
By combining the above equation with Equation (4), we get
By using the Cauchy product on the right-hand side of the above equation, we get
Making some elementary calculations in the above equation, after that equating coefficients of \(\frac{t^{n}}{n !}\) on both sides of the final equation, we arrive at the desired result. □
Replacing a by e in (13), we get the following corollary.
Corollary 8
If n is a nonnegative integer, then
Theorem 9
If n and k are nonnegative integers, then
Proof
We set the following functional equation:
Combining the above equation with (3) and (4), we get
Therefore
Making some elementary calculations in the above equation, after that equating the coefficients of \(\frac{t^{n}}{n!}\) on both sides of the final equation, we arrive at the desired result. □
5 Conclusion
Recently, many researchers have investigated and studied new families of special numbers and polynomials since these numbers and polynomials are used not only in science, but also in social sciences. That is, these numbers and polynomials have many applications in mathematics, in probability and statistics, in physics, in engineering, and in economic problems. Especially, polynomials have basic operations, which are addition, subtraction, multiplication, polynomials. Therefore, researchers can use them. In work of Simsek and Yardimci [12], we see that polynomials are used as models, related to differential equations, for real word problems related to sciences, approximate or curve fit experimental data, calculate beam deflection under loading, represent some properties of gases, and perform computer-aided geometric design in engineering. Polynomials represent the characteristics of a linear dynamic system, and we also know that a ratio of two polynomials represents a transfer function of a linear dynamic system. With the help of polynomials, one defines basis used in finite element computation and constructs parametric curves. Due to applications of the special numbers and polynomials with their generating functions, in this paper we investigate and study new families of special numbers and polynomials of higher order. We give their generating functions including their functional equations and partial derivative formulas.
By using these functions, we give many new identities, derivative formulas, and recurrence relations for these new families of special numbers and polynomials of higher order. These identities and relations will be potentially used in mathematics, in probability and statistics, in physics, in engineering, and in economic problems.
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Funding
This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2018R1D1A1B07041132).
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Simsek, Y., Kim, D. Identities and recurrence relations of special numbers and polynomials of higher order by analysis of their generating functions. J Inequal Appl 2018, 220 (2018). https://doi.org/10.1186/s13660-018-1815-7
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DOI: https://doi.org/10.1186/s13660-018-1815-7