Abstract
In this paper, we investigate degenerate versions of the generalized pth order Franel numbers which are certain finite sums involving powers of binomial coefficients. In more detail, we introduce degenerate generalized hypergeometric functions and study degenerate hypergeometric numbers of order p. These numbers involve powers of λ-binomial coefficients and λ-falling sequence, and can be represented by means of the degenerate generalized hypergeometric functions. We derive some explicit expressions and combinatorial identities for those numbers. We also consider several related special numbers like λ-hypergeometric numbers of order p and Apostol type λ-hypergeometric numbers of order p, of which the latter reduce in a limiting case to the generalized pth order Franel numbers.
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1 Introduction
First, we study certain finite sums involving powers of binomial coefficients which are called generalized pth order Franel numbers and can be represented in terms of hypergeometric functions. Then, among other things, we find that particular cases of these numbers are connected with many known special numbers and polynomials which include Bernoulli numbers, Euler numbers, Changhee numbers, Daehee numbers, Stirling numbers of the first kind, Catalan numbers, and Legendre polynomials.
In recent years, many mathematicians have devoted their attention to studying various degenerate versions of some special numbers and polynomials [4, 7, 8, 10, 15, 16]. The idea of investigating degenerate versions of some special numbers and polynomials originated from Carlitz’s papers [2, 3]. Indeed, he introduced the degenerate Bernoulli and Euler polynomials and numbers, and investigated some arithmetic and combinatorial aspects of them. Here we mention in passing that the degenerate Bernoulli polynomials were later rediscovered by Ustinov under the name of Korobov polynomials of the second kind. Two of the present authors, their colleagues, and some other people have studied quite a few degenerate versions of special numbers and polynomials with their interest not only in combinatorial and arithmetic properties but also in differential equations and certain symmetric identities (see [9, 16] and the references therein). It is worth noting that this idea of considering degenerate versions of some special polynomials and numbers is not only limited to polynomials but can also be extended to transcendental functions like gamma functions [11, 12]. We believe that studying some degenerate versions of special polynomials and numbers is a very fruitful and promising area of research in which many things remain yet to be uncovered.
Recently, Dolgy and Kim gave some explicit formulas of degenerate Stirling numbers associated with the degenerate special numbers and polynomials. Motivated by Dolgy and Kim’s paper [4], we would like to investigate degenerate versions of the generalized pth order Franel numbers. In more detail, we introduce degenerate generalized hypergeometric functions and study degenerate hypergeometric numbers of order p. These numbers involve powers of λ-binomial coefficients and λ-falling sequence, and can be represented by means of the degenerate generalized hypergeometric functions. We also consider several related special numbers like λ-hypergeometric numbers of order p and Apostol type λ-hypergeometric numbers of order p, of which the latter reduce in a limiting case to the generalized pth order Franel numbers.
For the rest of this section, we will fix some notations and recall some known results that are needed throughout this paper.
For \(\lambda\in\mathbb{R}\), the degenerate exponential function is defined as
From (1), we note that
where \((x)_{n,\lambda}\) is the λ-falling sequence given by
In [10], the degenerate Stirling numbers of the second kind are defined by
Let
Then \(\lim_{\lambda\rightarrow0}S_{2,\lambda}(n,k)=S_{2}(n,k) \), where \(S_{2}(n,k)\) are the ordinary Stirling numbers of the second kind given by
The Stirling numbers of the first kind are defined as
Thus, by (7), we get
In view of (4), the degenerate Stirling numbers of the first kind are defined by
Note that \(\lim_{\lambda\rightarrow0}S_{1,\lambda }(n,k)=S_{1}(n,k)\) (\(n,k\ge0\)).
As is well known, the generalized hypergeometric function \(F^{(p,q)}\) is defined by
where \(\langle a\rangle_{k}=a(a+1)\cdots(a+(k-1))\) (\(k\ge1\)), \(\langle a\rangle_{0}=1\) (see [17, 21]).
For example,
The Gauss summation theorem is given by
where \(R(c)>R(b)>0\), \(R(c-a-b)>0\), \(R(c)>R(a)>0\).
From (11), we note that
where \(R(c)>R(b)>0\).
The following are well-known identities related to the binomial coefficients:
2 Sums of powers of λ-binomial coefficients
The λ-binomial coefficients are defined as
From (18), we easily get
By (1), we easily get
where n and k are positive integers.
Note that \(\lim_{\lambda\rightarrow0}B_{\lambda}^{*}(n,k)=B(n,k)\), where \(B(n,k)\) are defined by Golombek and given by
Now, we define the degenerate hypergeometric function as
where \(\langle a\rangle_{n,\lambda}=a(a+\lambda)\cdots (a+(n-1)\lambda)\) (\(n\ge1\)), \(\langle a\rangle_{0,\lambda}=1\).
From (21), we note that
where n is a nonnegative integer.
Let us define
Therefore, by (22) and (23), we obtain the following theorem.
Theorem 2.1
For\(m\ge0\), we have
We note that \(\lim_{\lambda\rightarrow0}Q_{\lambda}(m,2)=\sum_{k=0}^{n}\binom{n}{k}^{2}k^{m}=Q(m,2) \), which was introduced by Golombek and Marburg (see [5, 6]).
We observe that
For \(n\in\mathbb{N}\), let
On the one hand, we have
On the other hand, we get
From (24), (25), and (26), we obtain the following theorem.
Theorem 2.2
For\(n\in\mathbb{N}\)and\(m\in\mathbb{N}\cup\{0\}\), we have
As is well known, the degenerate Bell polynomials are defined by
By (27), we easily get
Now, we define the degenerate bivariate Bell polynomials by
Thus, by (29), we get
From Theorem 2.2 and (30), we obtain the following corollary.
Corollary 2.3
For\(n\in\mathbb{N}\)and\(m\in\mathbb{N}\cup\{0\}\), we have
Note that
We observe that
Thus, by (31), we get
From (32), we have
For \(m=2\), we have
By (33), we get
Let us take \(m=3\). Then we have
Note that
and
For \(s\in\mathbb{C}\) with \(R(s)>0\), the gamma function is defined by
Let n be a nonnegative integer. Then
where \(R (\frac{c}{\lambda} )>0\) and \(R (\frac {b}{\lambda} )>0\).
For \(R (\frac{c}{\lambda} )>R (\frac{b}{\lambda} )>0\), we have
From (35), we note that
In particular, for \(z=\frac{1}{\lambda}\) (\(\lambda\ne0\)), from (11) we get
For \(n\in\mathbb{N}\), by (37), we get
where λ is a positive real number.
On the other hand,
Therefore, by (38) and (39), we obtain the following theorem.
Theorem 2.4
Letλbe a positive real number. For\(n\in\mathbb{N}\), we have
Note that
Now, we define the degenerate generalized hypergeometric function as
Let n be a positive integer. Then we define the degenerate hypergeometric numbers of orderp by
From (23) and (24), we note that \(H_{\lambda }(n,m)=H_{\lambda}^{(1)}(n,m)\), and \(Q_{\lambda}(m,2)=H_{\lambda }^{(2)}(n,m)\).
In (40), we note that
Therefore, by (41) and (42), we obtain the following theorem.
Theorem 2.5
For\(n,p\in\mathbb{N}\)and\(m\in\mathbb{N}\cup\{0\}\), we have
Note that
From (42), we note that
Therefore, by (41) and (43), we obtain the following theorem.
Theorem 2.6
For\(n,p\in\mathbb{N}\)and\(m\in\mathbb{N}\cup\{0\}\), we have
Note that
From (40), we have
Thus, by (44), we get
Note that
3 Further remarks
Let n be a positive integer. From (10), we have
Now, we define the λ-hypergeometric numbers of orderp by
where \(n\in\mathbb{N}\) and \(m\in\mathbb{N}\cup\{0\}\).
The alternatingλ-hypergeometric numbers of orderp are defined by
By (10), we get
where \(m\in\mathbb{N}\cup\{0\}\) and \(n\in\mathbb{N}\).
In general, we have
where \(n,p\in\mathbb{N}\) and \(m\in\mathbb{N}\cup\{0\}\).
We observe that
and
Thus, we note that
where \(m\in\mathbb{N}\cup\{0\}\) and \(n\in\mathbb{N}\).
For example,
where \(\delta_{n,k}\) is Kronecker’s symbol.
From (49), we note that
On the other hand,
Thus, by (54) and (55), we get
If \(m< n\), then \(T_{m,\lambda}^{(1)}(n)=0\).
Theorem 3.1
For\(n\in\mathbb{N}\)and\(m\in\mathbb{N}\cup\{0\}\), we have
In particular, if\(m< n\), then
For \(k\ge0\), we have
Corollary 3.2
For\(k\ge0\)and\(n\in\mathbb{N}\), we have
It is easy to show that
Thus, we have
For \(k\ge0\), \(n\in\mathbb{N}\), by (57), we get
From Corollary 3.2 and (58), we have
For \(\lambda,\lambda_{1}\in\mathbb{R}\), let us define Apostol type alternatingλ-hypergeometric numbers of orderp by
By (10), we get
On the other hand,
For \(k\ge0\) and \(n\in\mathbb{N}\), by (59), (60), and (61), we obtain the following theorem.
Theorem 3.3
For\(\lambda,\lambda_{1}\in\mathbb{R}\), \(n\in\mathbb{N}\), and\(k\in \mathbb{N}\cup\{0\}\), we have
Note that
For \(\lambda_{1}\in\mathbb{R}\), let us define Apstol–Stirling numbers of the second kind as
Now, we observe that
For \(k\in\mathbb{N}\cup\{0\}\) and \(n\in\mathbb{N}\), by (62) and (63), we get
From Theorem 3.3 and (64), we have
Therefore, by (65), we obtain the following corollary.
Corollary 3.4
For\(k\in\mathbb{N}\cup\{0\}\)and\(n\in\mathbb{N}\), we have
Remarks
-
(a)
Corollary 3.4 naturally interprets the sum in Corollary 3.4 in terms of Apostol–Stirling numbers of the second kind defined by (62).
-
(b)
For \(\lambda,\lambda_{1}\in\mathbb{R}\), let us define Apostol typeλ-hypergeometric numbers of orderp by
(66)By (10), we get
(67)$$ H_{m,\lambda}^{(p)}(n \mid\lambda_{1})=\sum _{k=0}^{n}\binom {n}{k}^{p} \lambda_{1}^{p}(k)_{m,\lambda},\quad\text{where } n,p\in \mathbb{N} \text{ and } m\in\mathbb{N}\cup\{0\}. $$Note that
$$ \lim_{\lambda\rightarrow0}H_{m,\lambda}^{(p)}(n\mid\lambda _{1})=\sum_{k=0}^{n} \binom{n}{k}^{p}\lambda_{1}^{p}k^{m}. $$
4 Conclusion
In this paper, we studied certain finite sums involving powers of binomial coefficients which are called generalized pth order Franel numbers and can be represented in terms of hypergeometric functions. Then, among other things, we found that particular cases of these numbers are connected with many known special numbers and polynomials which include Bernoulli numbers, Euler numbers, Changhee numbers, Daehee numbers, Stirling numbers of the first kind, Catalan numbers, and Legendre polynomials. Recently, Dolgy and Kim gave some explicit formulas of degenerate Stirling numbers associated with the degenerate special numbers and polynomials. Motivated by Dolgy and Kim’s paper [4], we investigated degenerate versions of the generalized pth order Franel numbers. In more detail, we introduced degenerate generalized hypergeometric functions and studied degenerate hypergeometric numbers of order p. We showed that the degenerate hypergeometric numbers of order p involve powers of λ-binomial coefficients and λ-falling sequence, and can be represented by means of the degenerate generalized hypergeometric functions. We also considered several related special numbers like λ-hypergeometric numbers of order p and Apostol type λ-hypergeometric numbers of order p, of which the latter reduce in a limiting case to the generalized pth order Franel numbers.
References
Agarwal, P., Younis, J.A., Kim, T.: Certain generating functions for the quadruple hypergeometric series \(K_{10}\). Notes Number Theory Discrete Math. 25(4), 16–23 (2019)
Carlitz, L.: A degenerate Staudt–Clausen theorem. Arch. Math. (Basel) 7, 28–33 (1956)
Carlitz, L.: Degenerate Stirling, Bernoulli and Eulerian numbers. Util. Math. 15, 51–88 (1979)
Dolgy, D.V., Kim, T.: Some explicit formulas of degenerate Stirling numbers associated with the degenerate special numbers and polynomials. Proc. Jangjeon Math. Soc. 21(2), 309–317 (2018)
Golombek, R.: Aufgabe 1088. Elem. Math. 49, 126–127 (1994)
Golombek, R., Marburg, D.: Aufgabe 1088, Summen mit Quadraten von Binomialkoeffizienten. Elem. Math. 50, 125–131 (1995)
Haroon, H., Khan, W.A.: Degenerate Bernoulli numbers and polynomials associated with degenerate Hermite polynomials. Commun. Korean Math. Soc. 33(2), 651–669 (2018)
He, Y., Araci, S.: Sums of products of Apostol–Bernoulli and Apostol–Euler polynomials. Adv. Differ. Equ. 2014, Article ID 155 (2014)
Kim, D.S., Kim, H.Y., Kim, D., Kim, T.: Identities of symmetry for type 2 Bernoulli and Euler polynomials. Symmetry 11, Article ID 613 (2019)
Kim, T.: A note on degenerate Stirling polynomials of the second kind. Proc. Jangjeon Math. Soc. 20(3), 319–331 (2017)
Kim, T., Jang, G.-W.: A note on degenerate gamma function and degenerate Stirling number of the second kind. Adv. Stud. Contemp. Math. (Kyungshang) 28(2), 207–214 (2018)
Kim, T., Kim, D.S.: Degenerate Laplace transform and degenerate gamma function. Russ. J. Math. Phys. 24(2), 241–248 (2017)
Kim, T., Kim, D.S.: A note on type 2 Changhee and Daehee polynomials. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 113(3), 2783–2791 (2019)
Kim, T., Kim, D.S., Kwon, J.: A note on λ-hypergeometric random variables. Rev. Educ. 387(2), 49–52 (2020)
Kim, T., Yao, Y., Kim, D.S., Jang, G.W.: Degenerate-Stirling numbers and r-Bell polynomials. Russ. J. Math. Phys. 25(1), 44–58 (2018)
Pyo, S.-S., Kim, T.: Some identities of fully degenerate Bell polynomials arising from differential equations. Proc. Jangjeon Math. Soc. 22(2), 357–363 (2019)
Rainville, E.D.: Special Functions. Chelsea, New York (1971)
Rim, S.-H., Kim, T., Lee, S.H.: Some symmetry identities for \((h,q)\)-Bernoulli polynomials under the third dihedral group \(D_{3}\) arising from q-Volkenborn integral on \(\mathbb{Z}_{p}\). J. Comput. Anal. Appl. 20(3), 432–436 (2016)
Rim, S.-H., Kim, T., Pyo, S.-S.: Identities between harmonic, hyperharmonic and Daehee numbers. J. Inequal. Appl. 2018, Article ID 168 (2018)
Roman, S.: The Umbral Calculus. Pure and Applied Mathematics, vol. 111. Academic Press, New York (1984)
Whittaker, E.T., Watson, G.N.: Modern Analysis, 4th edn. Cambridge University Press, Cambridge (1927)
Zhang, W., Lin, X.: Identities involving trigonometric functions and Bernoulli numbers. Appl. Math. Comput. 334, 288–294 (2018)
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The authors would like to thank the anonymous reviewers for their valuable comments and suggestions which helped us improve the quality of our work.
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This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) (No. 2017R1E1A1A03070882).
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Kim, T., Kim, D.S., Lee, H. et al. Degenerate binomial coefficients and degenerate hypergeometric functions. Adv Differ Equ 2020, 115 (2020). https://doi.org/10.1186/s13662-020-02575-3
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DOI: https://doi.org/10.1186/s13662-020-02575-3