Abstract
The aim of this paper is to study probabilistic versions of the degenerate Stirling numbers of the second kind and the degenerate Bell polynomials, namely the probabilisitc degenerate Stirling numbers of the second kind associated with \(Y\) and the probabilistic degenerate Bell polynomials associated with \(Y\), which are also degenerate versions of the probabilisitc Stirling numbers of the second and the probabilistic Bell polynomials considered earlier. Here \(Y\) is a random variable whose moment generating function exists in some neighborhood of the origin. We derive some properties, explicit expressions, certain identities and recurrence relations for those numbers and polynomials. In addition, we treat the special cases that \(Y\) is the Poisson random variable with parameter \(\alpha (>0)\) and the Bernoulli random variable with probability of success \(p\).
DOI 10.1134/S106192082304009X
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1. Introduction
It is amusing to witness that various degenerate versions of many special numbers and polynomials have been studied recently not only with their number-theoretic or combinatorial interests but also with their applications to others, including probability, quantum mechanics and differential equations. This exploration for degenerate versions began with the work by Carlitz on the degenerate Bernoulli and degenerate Euler polynomials in [4]. It is remarkable that many different tools, like generating functions, combinatorial methods, \(p\)-adic analysis, umbral calculus, operator theory, differential equations, special functions, probability theory and analytic number theory, are employed in the course of this quest (see [5, 9, 11-19] and the references therein).
Let \(Y\) be a random variable satisfying the moment conditions (see (15)). The aim of this paper is to introduce and study probabilistic versions of the degenerate Stirling numbers of the second kind and the degenerate Bell polynomials, namely the probabilisitc degenerate Stirling numbers of the second kind associated with \(Y\) and the probabilistic degenerate Bell polynomials associated with \(Y\), which are also degenerate versions of the probabilisitc Stirling numbers of the second and the probabilistic Bell polynomials considered in [2]. The definitions for those numbers and polynomials in (20) and (23) are very natural, as one can easily figure out. Then we derive some properties, explicit expressions, certain identities and recurrence relations for those numbers and polynomials. In addition, we consider the special cases that \(Y\) is the Poisson random variable with parameter \(\alpha (>0)\) and the Bernoulli random variable with probability of success \(p\), and derive several identities.
The outline of this paper is as follows. In Section 1, we recall the degenerate exponentials and degenerate logarithms. We remind the reader of the partial and complete Bell polynomials. We recall the degenerate Stirling polynomials and numbers of the second kind together with their explicit expressions. We also remind the reader of the degenerate Bell polynomials \(\phi_{n,\lambda}(x)\). Assume that \(Y\) is a random variable such that the moment generating function of \(Y\), \(E[e^{tY}]=\sum_{n=0}^{\infty}\frac{t^{n}}{n!}E[Y^{n}], \quad (|t| <r)\), exists for some \(r >0\). Let \((Y_{j})_{j\ge 1}\) be a sequence of mutually independent copies of the random variable \(Y\), and let \(S_{k}=Y_{1}+Y_{2}+\cdots+Y_{k},\,\, (k \ge 1)\), with \(S_{0}=0\). Then we recall the probablistic Stirling numbers of the second kind associated with \(Y\), \(S_{Y}(n,m)\), which are defined in terms of the \(n\)th moments of \(S_{k},\,\,(k=0,1,\dots,m)\), (see [2]). Section 2 is the main result of this paper. Let \((Y_{j})_{j \ge1},\,\, S_{k},\,\, (k=0,1,\dots)\) be as in the above. Then we first define the probabilistic degenerate Stirling numbers of the second kind associated with the random variable \(Y\), \({n \brace k}_{Y,\lambda}\), as a degenerate version of \(S_{Y}(n,k)\). We derive for \({n \brace k}_{Y,\lambda}\) an explicit expression in Theorem 2.1 and an expression in term of the partial Bell polynomial in Theorem 2.10. Next, we define the probabilistic degenerate Bell polynomials associated with the random variable \(Y\), \(\phi_{n,\lambda}^{Y}(x)\), as a natural extension of the numbers \({n \brace k}_{Y,\lambda}\). Then we derive for \(\phi_{n,\lambda}^{Y}(x)\) a Dobinski-like formula in Theorem 2.2 and an expression in terms of the partial Bell polynomials in Theorem 2.3. We obtain for \(\phi_{n,\lambda}^{Y}(x)\) a recurrence formula in Theorem 2.4 and a binomial identity in Theorem 2.5. In Theorem 2.6, three identities all related to the partial Bell polynomials are obtained. An expression for the \(k\)th derivative of \(\phi_{n,\lambda}^{Y}(x)\) is derived in Theorem 2.7. In Theorems 2.8 and 2.9, some identities involving the degenerate Stirling polynomials of the second kind and the generalized falling factorials are obtained. Let \(Y\) be the Poisson random variable with parameter \(\alpha (>0)\). Then we derive an identity involving \(\phi_{n,\lambda}(k\alpha)\) and \({m \brace k}_{Y,\lambda},\,\,(k=0,1,\dots,m)\), in Theorem 2.11 and an expression of \(\phi_{n,\lambda}^{Y}(x)\) in terms of \({n \brace k}_{\lambda}\) and \(\phi_{k}(x)\) in Theorem 2.12. Let \(Y\) be the Bernoulli random variable with probability of success \(p\). Then we show \(\phi_{n,\lambda}^{Y}(x)=\phi_{n,\lambda}(px)\) in Theorem 2.13. Also, we find an integral representation for the finite sum \(\frac{1}{k+1}\Big((1)_{n,\lambda}+(2)_{n,\lambda}+\cdots+(k)_{n,\lambda}\Big)\) in Theorem 2.14.
For any nonzero \(\lambda\in\mathbb{R}\), the degenerate exponentials are defined by
where the generalized falling factorials are given by
For \(x=1\), we write \(e_{\lambda}(t)=e_{\lambda}^{1}(t)\). Let \(\log_{\lambda}(t)\) be the degenerate logarithm which is the compositional inverse of \(e_{\lambda}(t)\) satisfying
Then we have
For any integer \(k \ge 0\), the partial Bell polynomials are given by
where
The complete Bell polynomials are defined by
The Stirling numbers of the second are given by
From (5) and (6), we note that
and
where \(\phi_{n}(x)\) are the Bell polynomials given by
The degenerate Stirling numbers of the second kind are defined by
From (9), we note that
Thus, by (10), we easily get
Note that \(\lim_{\lambda\rightarrow 0}{n \brace k}_{\lambda}={n \brace k},\ (n,k\ge 0)\). In [11], the degenerate Stirling polynomials of the second kind are introduced by Kim as
Note that \({n \brace k}_{\lambda}=S_{2,\lambda}(n,k|0),\ (n,k\ge 0)\). From (12), we have
Recently, Kim-Kim introduced the degenerate Bell polynomials given by
Note that \(\lim_{\lambda\rightarrow 0}\phi_{n,\lambda}(x)=\phi_{n}(x),\ (n\ge 0)\).
We assume that \(Y\) is a random variable satisfying the moment conditions
for some \(r>0\), where \(E\) stands for the mathematical expectation. The equation (15) guarantees the existence of the moment generating function of \(Y\) given by
Let \((Y_{j})_{j\ge 1}\) be a sequence of mutually independent copies of the random variable \(Y\), and let
The probabilistic Stirling numbers of the second kind associated with the random variable \(Y\) are defined by
2. Probabilistic degenerate Bell polynomials associated with random variables
Let \((Y_{j})_{j\ge 1}\) be a sequence of mutually independent copies of the random variable \(Y\), and let
Now, we consider the probabilistic degenerate Stirling numbers of the second kind associated with the random variable \(Y\) given by
On the other hand, by (1), we get
Thus, by (20) and (21), we get
By (11), we note that if \(Y=1\), then \({n \brace k}_{Y,\lambda}={n \brace k}_{\lambda}\). Therefore, by (22), we obtain the following theorem.
Theorem 2.1.
For \(n,k\) with \(n\ge k \ge 0\), we have
We define the probabilistic degenerate Bell polynomials associated with the random variable \(Y\) by
Note that if \(Y=1\), then \(\phi_{n,\lambda}^{Y}(x)=\phi_{n,\lambda}(x),\ (n\ge 0)\). For \(x=1,\ \phi_{n,\lambda}^{Y}=\phi_{n,\lambda}(1)\) are called the probabilistic degenerate Bell numbers associated with the random variable \(Y\). From (23), we note that
By (24), we get
Therefore, by comparing the coefficients on both sides of (25), we obtain the following Dobinski-like formula.
Theorem 2.2.
For \(n\ge 0\), we have
From, (24), we have
Therefore, by comparing the coefficients on both sides of (26), we obtain the following theorem.
Theorem 2.3.
For \(n\ge 0\), we have
From (24), we have
Therefore, by comparing the coefficients on both sides of (27), we obtain the following recurrence relation.
Theorem 2.4.
For \(n\ge 0\), we have
Now, we observe that
Therefore, by (28), we obtain the following binomial identity.
Theorem 2.5.
For \(n\ge 0\), we have
From (24), we note that
Thus, by comparing the coefficients on both sides of (29), we get
Now, we observe that
By (31), we get
From (4) and (32), we note that
Thus, by (33), we get
From (4), we note taht
By comparing the coefficients on both sides of (35), we have
where \(n\ge k\ge 0\). Therefore, by (30), (34), and (36), we obtain the following theorem.
Theorem 2.6.
The following identities hold true.
and
From (24), we note that
Thus, by (37), we get
In particular, for \(k=1\), we have
Thus, by comparing the coefficients on both sides of (39), we get
Therefore, by (38) and (40), we obtain the following theorem.
Theorem 2.7.
For \(n,k\in\mathbb{N}\) with \(n\ge k\), we have
In particular, for \(k=1\), we have
Let \(f\) be a real valued function. The forward difference operator \(\triangle\) is defined by \(\triangle f(x)=f(x+1)-f(x)\). For \(y\in\mathbb{R}\), the operator \(\triangle_{y}\) is defined by
together with the iterates
From (41) and (42), we note that
Thus, by (42), we get
where
From (44), we note that
where \(m\) is a positive integer.
By inversion, from (45), we have
From (42), we note that
where \(N\) is a nonnegative integer. Thus, by (13) and (43), we have
where \(n,k\) are nonnegative integers with \(n\ge k\).
From (47) and (48), we note that
The equation (49) can be rewritten as
Therefore, by (50), we obtain the following theorem.
Theorem 2.8.
For \(N\ge 0\), we have
where
In view of (12), we define the probabilistic degenerate Stirling polynomials of the second kind associated with the random variable \(Y\) by
On the other hand, by binomial expansion, we get
From (45), we note that
Therefore, by (56), we obtain the following theorem.
Theorem 2.9.
For \(m,n\ge 0\), we have
The Cauchy numbers of order \(r\) are given by
Let \(Y\) be a continuous uniform random variable on (0,1), and let \((Y_{j})_{j\ge 1}\) be a sequence of mutually independent copies of the random variable \(Y\). Then we have
Thus, by (57), we get
By comparing the coefficients on both sides of (58), we get
where \(n\ge k\ge 0\).
where \(m\) is a nonnegative integer. Then, from (59) and (60), we have
From (20), we note that
Thus, we get
where \(n\ge k\ge 0\). Therefore, by (63), we obtain the following theorem.
Theorem 2.10.
For \(n\ge k\ge 0\), we have
Let \(Y\) be the Poisson random variable with parameter \(\alpha(>0)\). Then we have
By comparing the coefficients on both sides (64), we get
Therefore, by (56) and (65), we obtain the following theorem.
Theorem 2.11.
Let \(Y\) be the Poisson random variable with parameter \(\alpha(>0)\). Then we have
Let \(Y\) be the Poisson random variable with parameter \(\alpha(>0)\). Then, from (24), we note that
Therefore, by (66), we obtain the following theorem.
Theorem 2.12.
Let \(Y\) be the Poisson random variable with parameter \(\alpha(>0)\). Then we have
Let \(Y\) be the Bernoulli random variable with probability of success \(p\). Then we have
Thus, by (63) and (67), we get
where \(n\ge k\ge 0\). From (68), we have
Therefore, by (68) and (69), we obtain the following theorem.
Theorem 2.13.
Let \(Y\) be the Bernoulli random variable with probability of success \(p\). Then we have
In particular, for \(n\ge k\ge 0\), we get
We observe that, for \(n,k \ge 1\), we have
By (70), we get
Let \(Y\) be the Bernoulli random variable with probability of success \(p\), and let \((Y_{j})_{j\ge 1}\) be mutually independent copies of \(Y\). Then we have
Thus, by (72), we get
From (46) and (73), we note that
For \(n\in\mathbb{N}\), by (74), we get
Thus, we see that
Therefore, by (77), we obtain the following theorem.
Theorem 2.14.
Let \(Y\) be the Bernoulli random variable with probability of success \(p\). For \(n,k\in\mathbb{N}\), we have
3. conclusion
In this paper, we studied by using generating functions probabilistic degenerate Stirling numbers of the second associated with \(Y\) and the probabilistic degenerate Bell polynomials associated with \(Y\) as degenerate versions of the probabilistic Stirling numbers of the second associated with \(Y\) and probabilistic Bell polynomials associated with \(Y\), respectively (see [2]). Here \(Y\) is a random variable satisfying moment conditions in (15) so that the moment generating function of \(Y\) exists. In more detail, we derived some expressions and related identities for \({n \brace k}_{Y,\lambda}\) (see Theorems 2.1, 2.6, 2.10), an identity involving \(S_{Y,\lambda}(n,k|x)\) (see Theorem 2.9), some expressions for \(\phi_{n,\lambda}^{Y}(x)\) (see Theorems 2.2, 2.3, 2.6), a recurrence relation (see Theorem 2.4) and some properties (see 2.5, 2.7) for \(\phi_{n,\lambda}^{Y}(x)\), an identity (see Theorem 2.11) and an expression for \(\phi_{n,\lambda}^{Y}(x)\) (see Theorem 2.12) when \(Y\) is a Poisson random variable with parameter \(\alpha (>0)\), and an expression for \(\phi_{n,\lambda}^{Y}(x)\) (see Theorem 2.13) and an integral representation for \(\frac{1}{k+1}\Big((1)_{n,\lambda}+(2)_{n,\lambda}+\cdots+(k)_{n,\lambda}\Big)\) (see Theorem 2.14) when \(Y\) is a Bernoulli random variable with probability of success \(p\).
As one of our future projects, we would like to continue to study degenerate versions, \(\lambda\)-analogues and probabilistic versions of many special polynomials and numbers and to find their applications to physics, science and engineering as well as to mathematics.
References
M. Abbas and S. Bouroubi, “On New Identities for Bell’s Polynomials”, Discrete Math., 293:1–3 (2005), 5–10.
José A. Adell, “Probabilistic Stirling Numbers of the Second Kind and Applications”, J. Theoret. Probab., 35:1 (2022), 636–652.
K. Boubellouta, A. Boussayoud, S. Araci, and M. Kerada, “Some Theorems on Generating Functions and Their Applications”, Adv. Stud. Contemp. Math., Kyungshang, 30:3 (2020), 307–324.
L. Carlitz, “Degenerate Stirling, Bernoulli and Eulerian numbers”, Utilitas Math., 15 (1979), 51–88.
S.-K. Chung, G.-W. Jang, J. Kwon, and J. Lee, “Some Identities of the Degenerate Changhee Numbers of Second Kind Arising from Differential Equations”, Adv. Stud. Contemp. Math., Kyungshang, 28:4 (2018), 577–587.
L. Comtet, Advanced combinatorics. The art of finite and infinite expansions, Revised and enlarged edition, D. Reidel Publishing Co., Dordrecht, 1974.
D. Gun and Y. Simsek, “Combinatorial Sums Involving Stirling, Fubini, Bernoulli Numbers and Approximate Values of Catalan Numbers”, Adv. Stud. Contemp. Math., Kyungshang, 30:4 (2020), 503–513.
D. S. Kim and T. Kim, “Normal Ordering Associated with \(\lambda\)-Whitney Numbers of the First Kind in \(\lambda\)-Shift Algebra”, Russ. J. Math. Phys., 30:3 (2023), 310–319.
D. S. Kim and T. Kim, “A Note on a New Type of Degenerate Bernoulli Numbers”, Russ. J. Math. Phys., 27:2 (2020), 227–235.
D. S. Kim, T. Kim, S.-H. Lee, and J.-W. Park, “Some New Formulas of Complete and Incomplete Degenerate Bell Polynomials”, Adv. Difference Equ., :326 (2021).
T. Kim, “A Note on Degenerate Stirling Polynomials of the Second Kind”, Proc. Jangjeon Math. Soc., 20:3 (2017), 319–331.
T. Kim and D. S. Kim, “Some Identities on Truncated Polynomials Associated with Degenerate Bell Polynomials”, Russ. J. Math. Phys., 28:3 (2021), 342–355.
T. Kim and D. S. Kim, “Degenerate \(r\)-Whitney Numbers and Degenerate \(r\)-Dowling Polynomials via Boson Operators”, Adv. in Appl. Math., 140:102394 (2022).
T. Kim and D. S. Kim, “Degenerate Zero-Truncated Poisson Random Variables”, Russ. J. Math. Phys., 28:1 (2021), 66–72.
T. Kim and D. S. Kim, “Some Identities Involving Degenerate Stirling Numbers Associated with Several Degenerate Polynomials and Numbers”, Russ. J. Math. Phys., 30:1 (2023), 62–75.
T. Kim, D. S. Kim, and D. V. Dolgy, “On Partially Degenerate Bell Numbers and Polynomials”, Proc. Jangjeon Math. Soc., 20:3 (2017), 337–345.
T. Kim, D. S. Kim, D. V. Dolgy, and J.-W. Park, “Degenerate Binomial and Poisson Random Variables Associated with Degenerate Lah-Bell Polynomials”, Open Math., 19:1 (2021), 1588–1597.
T. Kim, D. S. Kim, J. Kwon, H. Lee, and S.-H. Park, “Some Properties of Degenerate Complete and Partial Bell Polynomials”, Adv. Difference Equ., :304 (2021).
S.-S. Pyo, “Degenerate Cauchy Numbers and Polynomials of the Fourth Kind”, Adv. Stud. Contemp. Math., Kyungshang, 28:1 (2018), 127–138.
S. Roman, The Umbral Calculus, Pure and Applied Mathematics 111, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1984.
S. M. Ross, Introduction to Probability Models, Twelfth edition of Academic Press, London, 2019.
R. Soni, P. Vellaisamy, and A. K. Pathak, “A Probabilistic Generalization of the Bell Polynomials”, J. Anal., (2023).
B. Q. Ta, “Probabilistic Approach to Appell Polynomials”, Expo. Math., 33:3 (2015), 269–294.
H. Teicher, “An Inequality on Poisson Probabilities”, Ann. Math. Statist., 26 (1955), 147–149.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
The authors of this work declare that they have no conflicts of interest.
Additional information
Publisher's Note. Pleiades Publishing remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Kim, T., Kim, D.S. Probabilistic Degenerate Bell Polynomials Associated with Random Variables. Russ. J. Math. Phys. 30, 528–542 (2023). https://doi.org/10.1134/S106192082304009X
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S106192082304009X