Abstract
Let \(Y\) be a random variable whose moment generating function exists in a neighborhood of the origin. The aim of this paper is to introduce and study the probabilistic extension of Bernoulli polynomials and Euler polynomials, namely the probabilistic Bernoulli polynomials associated \(Y\) and the probabilistic Euler polynomials associated with \(Y\). Also, we introduce the probabilistic \(r\)-Stirling numbers of the second associated \(Y\), the probabilistic two variable Fubini polynomials associated \(Y\), and the probabilistic poly-Bernoulli polynomials associated with \(Y\). We obtain some properties, explicit expressions, certain identities and recurrence relations for those polynomials. As special cases of \(Y\), we treat the gamma random variable with parameters \(\alpha,\beta > 0\), the Poisson random variable with parameter \(\alpha >0\), and the Bernoulli random variable with probability of success \(p\).
DOI 10.1134/S106192084010072
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1. Introduction
In recent years, we have witnessed that degenerate versions and \(\lambda\)-analogues of many special polynomials and numbers were investigated by employing various methods such as generating functions, combinatorial methods, umbral calculus, \(p\)-adic analysis, differential equations, probability, special functions, analytic number theory and operator theory (see [6, 11–18, 21] and the references therein). Here we study by means of generating functions probabilistic extensions of several special polynomials, including the Bernoulli and Euler polynomials.
Let \(Y\) be a random variable satisfying the moment condition (see (8)). The aim of this paper is to study, as probabilistic extensions, the probabilistic Bernoulli polynomials associated \(Y\) and the probabilistic Euler polynomials associated with \(Y\), along with the probabilistic \(r\)-Stirling numbers of the second associated \(Y\), the probabilistic two variable Fubini polynomials associated \(Y\), and the probabilistic poly-Bernoulli polynomials associated with \(Y\). We derive some properties, explicit expressions, certain identities and recurrence relations for those polynomials and numbers. In addition, as special cases of \(Y\), we consider the gamma random variable with parameters \(\alpha,\beta > 0\), the Poisson random variable with parameter \(\alpha >0\), and the Bernoulli random variable with probability of success \(p\).
The outline of this paper is as follows. In Section 1, we recall the Bernoulli polynomials, the Euler polynomials, the Stirling numbers of the second kind, the \(r\)-Stirling numbers of the second kind, the Fubini polynomials and two variable Fubini polynomials. Assume that \(Y\) is a random variable such that the moment generating function of \(Y\),
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
Then we remind the reader of the gamma random variable with parameters \(\alpha,\,\beta >0\) and the probabilistic Stirling numbers of the second. Section 2 is the main results of this paper. Let \((Y_{j})_{j \ge1},\,\, S_{k}\,\, (k=0,1,\dots)\) be as in the above. We define the probabilistic Bernoulli polynomials associated \(Y\), \(B_{n}^{Y}(x)\) (see (12)). We derive explicit expressions for \(B_{n}^{Y}(x)\) in Theorems 2.1, 2.2, and 2.6, and \(B_{n}^{Y}(x)=n!\binom{x+n-2}{n}\) in Theorem 2.9, when \(Y \sim \Gamma(1,1)\) (see (9)). In Theorems 2.3, 2.4, and 2.5, we get probabilistic analogues for the well-known identities
and
respectively. In Theorem 2.8, we deduce an identity involving \(B_{n}^{Y}=B_{n}^{Y}(0)\), which gives an explicit expression for the Bernoulli numbers
for \(Y=1\). We show that \(B_{n}^{Y}=\frac{1}{p}B_{n}\) if \(Y\) is the Bernoulli random variable with probability of success \(p\) in Theorem 2.10. We define the probabilistic \(r\)-Stirling numbers of the second kind associated with \(Y\) (see (31)), \({n+r \brace k+r}_{r,Y}\), for which an expression in terms of \(E[S_{j+r}^{n}]\) is found in Theorem 2.11. We introduce the probabilistic two variable Fubini polynomials associated \(Y\), \(F_{n}^{Y}(x|y)\) (see (33)). We obtain an expression in terms of \(F_{k}^{Y}(x)\) for \(F_{n}^{Y}(x|y)\) in Theorem 2.12. In case \(y=r\) is a nonnegative integer, we show that
in Theorem 2.13. In Theorem 2.14, we get an identity involving \(B_{n}^{Y}(r)\), which reduces to the identity
for \(Y=1\). We define the probabilistic poly-Bernoulli polynomials \(B_{n}^{(k,Y)}(x)\) (see (40)) by making use of the polylogarithmic function. In Theorem 2.15, we obtain an expression for \(B_{n}^{(k,Y)}(x)\) in terms of \(B_{k}^{Y}(x)\). We define the probabilistic Euler polynomials associated with \(Y\), \(\mathcal{E}_{n}^{Y}(x)\) (see (43)). We get an expression for \(\mathcal{E}_{n}^{Y}(x)\) in Theorem 2.16 and that for for \(\mathcal{E}_{n}^{Y}=\mathcal{E}_{n}^{Y}(0)\) in Theorem 2.17. In Theorem 2.18, we get an identity which corresponds to the well-known identity
for any integer \(m \ge 0\) and any even positive integer \(n\). We show that in Theorem 2.19, \(\mathcal{E}_{n}^{Y}=\frac{n!}{2^n}\), for \(Y \sim \Gamma(1,1)\) and in Theorem 2.20,
when \(Y\) is the Poisson random variable with parameter \(\alpha >0\). For the rest of this section, we recall the facts that are needed throughout this paper.
It is well known that the Bernoulli polynomials are defined by
When \(x=0\), \(B_{n}=B_{n}(0)\) are called the Bernoulli numbers. The Euler polynomials are given by
For \(x=0\), \(\mathcal{E}_{n}=\mathcal{E}_{n}(0)\) are called the Euler numbers.
For \(n\ge 0\), the Stirling numbers of the second kind are defined by
where \((x)_{0}=1,\ (x)_{n}=x(x-1)\cdots(x-n+1),\ (n\ge 1)\).
For \(r\in\mathbb{Z}\) with \(r\ge 0\), the \(r\)-Stirling numbers of the second kind are given by
If \(r=0,\ {n+r \brace k+r}_{r}={n\brace k},\ (n\ge k\ge 0)\).
The Fubini polynomials are defined by
Two variable Fubini polynomials are given by
For \(y=r\in\mathbb{Z}\) with \(r\ge 0\), we have
Assume that \(Y\) is a random variable such that the moment generating function of \(Y\),
Let \((Y_{j})_{j\ge 1}\) be a sequence of mutually independent copies of the random variable \(Y\), and let \(S_{k}=Y_{1}+Y_{2}+\cdots+Y_{k},\ (k\ge 1)\) and \(S_{0}=0\).
A continuous random variable \(Y\) whose density function is defined by
for some \(\alpha,\,\beta>0\) is said to be the gamma random variable with parameters \(\alpha,\beta\), which is denoted by \(Y\sim\Gamma(\alpha,\beta)\), (see [20, 24, 26–28]). The probabilistic Stirling numbers of the second kind associated with \(Y\) are given by
When \(Y=1\), we have \({n\brace k}_{Y}={n\brace k}\).
2. Probabilistic Bernoulli and Euler polynomials associate with random variables
Let \((Y_{j})_{j\ge 1}\) be a sequence of mutually independent copies of the random variable \(Y\), and let
We consider the probabilistic Bernoulli polynomials associated with \(Y\) which are given by
When \(Y=1\), we have \(B_{n}^{Y}(x)=B_{n}(x),\ (n\ge 0)\). For \(x=0,\ B_{n}^{Y}=B_{n}^{Y}(0)\) are called the probabilistic Bernoulli numbers associated with \(Y\).
From (12), we note that
Thus, by (13), we get
Therefore, by comparing the coefficients on both sides of (14), we obtain the following theorem.
Theorem 2.1.
Let \(n\) be a positive integer. Then we have
By binomial expansion, we get
Thus, by (12) and (15), we get
Therefore, by comparing the coefficients on both sides of (16), we obtain the following theorem.
Theorem 2.2.
For \(n\ge 0,\) we have
From (12), we note that
On the other hand, by (11), we get
Therefore, by (17) and (18), we obtain the following theorem.
Theorem 2.3.
For \(n,m\ge 0\), we have
By (12), we see
By comparing the coefficients on both sides of (18), we have
Therefore, by (20), we obtain the following theorem.
Theorem 2.4.
Let \(n\) be a nonnegative integer. Then we have
where \(\delta_{n,k}\) is the Kronecker’s symbol.
Now, we observe that
where \(d\) is a positive integer.
By comparing the coefficients on both sides of (21), we obtain the following theorem.
Theorem 2.5.
Let \(d\) be a positive integer. For \(n\ge 0,\) we have
In particular, for \(Y=1\), we have
Let \(Y\) be the Poisson random variable with parameter \(\alpha>0\). Then we have
Therefore, by (22), we obtain the following theorem.
Theorem 2.6.
Let \(Y\) be the Poisson random variable with parameter \(\alpha>0\). For \(n\ge 0,\) we have
We need the following lemma in showing Theorems 2.8 and 2.14. We obtain this from the following observation:
Lemma 1.
For \(n\in\mathbb{N},\) we have
Equivalently, for \(n\in\mathbb{N}\cup\{0\},\) we have
The probabilistic Fubini polynomials are given by
Thus, by (10), we get
From (24), we get
From (23), we have
Therefore, by (25) and (26) and using Lemma 1, we obtain the following theorem.
Theorem 2.7.
For \(n\ge 0,\) we have
In particular, for \(Y=1,\) we get
Let \(Y\sim\Gamma(1,1)\). Then we have
Thus, by (12) and (27), we get
Therefore, by (28), we obtain the following theorem.
Theorem 2.8.
Let \(Y\sim\Gamma(1,1)\). For \(n\ge 0,\) we have
Let \(Y\) be the Bernoulli random variable with the probability of success \(p\). Then we have
Therefore, by (30), we obtain the following theorem.
Theorem 2.9.
Let \(Y\) be the Bernoulli random variable with probability of success \(p\). For \(n\ge 0\), we have
Now, we define the probabilistic \(r\)-Stirling numbers of the second kind associated with \(Y\) by
where \(r\) is a nonnegative integer.
When \(Y=1\), we have \({n+r \brace k+r}_{r,Y}={n+r \brace k+r}_{r}\). From (31), we note that
Therefore, by (32), we obtain the following theorem.
Theorem 2.10.
For \(n\ge k\ge 0,\) we have
Now, we consider the probabilistic two variable Fubini polynomials associated with \(Y\) defined by
When \(Y=1\), we have \(F_{n}^{Y}(x|y)=F_{n}(x|y)\).
From (33), we note that
Therefore, by comparing the coefficients on both sides of (34), we obtain the following theorem.
Theorem 2.11.
For \(n\ge 0,\) we have
For \(r\in\mathbb{Z}\) with \(r\ge 0\) and from (31), we have
Therefore, by (35), we obtain the following theorem.
Theorem 2.12.
Let \(r\) be a nonnegative integer. For \(n\ge 0,\) we have
From (36), we have
By (37), we get
Therefore, by (37) and (38) and using Lemma 1, we obtain the following theorem.
Theorem 2.13.
For \(n,r\in\mathbb{Z}\) with \(n,r\ge 0,\) we have
In particular, for \(Y=1,\) we get
For \(k\in\mathbb{Z}\), the polylogarithmic function is given by
Note that \(\mathrm{Li}_{1}(x)=-\log(1-x)\).
Now, we define the probabilistic poly-Bernoulli polynomials associated with \(Y\) by
Note that
From (40), we note that
Therefore, by (42), we obtain the following theorem.
Theorem 2.14.
For \(n\ge 0,\) we have
Now, we define the probabilistic Euler polynomials associated with \(Y\) by
When \(Y=1\), \(\mathcal{E}_{n}^{Y}(x)=\mathcal{E}_{n}(x),\ (n\ge 0)\). In particular, for \(x=0\), \(\mathcal{E}_{n}^{Y}=\mathcal{E}_{n}^{Y}(0)\) are called the probabilistic Euler numbers associated with \(Y\).
From (43), we have
Therefore, by comparing the coefficients on both sides of (44), we obtain the following theorem.
Theorem 2.15.
For \(n\ge 0,\) we have
From (43), we note that
Therefore, by (46), we obtain the following theorem.
Theorem 2.16.
For \(n\ge 0,\) we have
For \(n\in\mathbb{N}\) with \(n\equiv 0\ (\mathrm{mod}\ 2)\), we have
On the other hand, by (11), we get
Therefore, by (47) and (48), we obtain the following theorem.
Theorem 2.17.
For \(n\in\mathbb{N}\) with \(n\equiv 0\ (\mathrm{mod}\ 2)\) and \(m\ge 0,\) we have
Let \(Y\sim\Gamma(1,1)\). Then, by (27), \(E[e^{Yt}] =\frac{1}{1-t},\quad (t <1)\). By (43), we get
Hence, by (49), we get
Theorem 2.18.
Let \(Y\sim\Gamma(1,1)\). For \(n\ge 1,\) we have
Let \(Y\) be the Poisson random variable with parameter \(\alpha>0\). Then we have
Therefore, by (51), we obtain the following theorem.
Theorem 2.19.
Let \(Y\) be the Poisson random variable with parameter \(\alpha>0\). For \(n\ge 0,\) we have
3. Conclusion
We used generating functions to study probabilistic extensions of several special polynomials, namely the probabilistic Bernoulli polynomials associated \(Y\) and the probabilistic Euler polynomials associated \(Y\), together with the probabilistic \(r\)-Stirling numbers of the second associated \(Y\), the probabilistic two variable Fubini polynomials associated \(Y\), and the probabilistic poly-Bernoulli polynomials associated with \(Y\). Here \(Y\) is a random variable such that the moment generating function of \(Y\) exists in a neighborhood of the origin. In more detail, we obtained several explicit expressions for \(B_{n}^{Y}(x)\) (see Theorems 2.1, 2.2, 2.6) and an explicit expression for each of \(F_{n}^{Y}(x|y),\,F_{n}^{Y}(y|r),\,B_{n}^{(k,Y)}(x)\), and \(\mathcal{E}_{n}^{Y}(x)\) (see Theorems 2.12, 2.13, 2.15, 2.16). We derived three identities about probabilistic extensions of some well-known identities on Bernoulli numbers (see Theorems 2.3–2.5) and a probabilistic extension of the identity that reduces to an explicit expression of those numbers (see Theorem 2.8). We got two identities on \(\mathcal{E}_{n}^{Y}\) (see Theorems 2.17, 2.18). In the special case of \(Y \sim \Gamma(1,1)\), we found an expression for \(B_{n}^{Y}(x)\) (see Theorem 2.9) and that for \(\mathcal{E}_{n}^{Y}\) (see Theorem 2.19). Also, we determined \(B_{n}^{Y}\) when \(Y\) is the Bernoulli random variable with probability of success \(p\) (see Theorem 2.10) and \(\mathcal{E}_{n}^{Y}\) when \(Y\) is the Poisson random variable with parameter \(\alpha\) (see Theorem 2.20). Finally, we deduced two identities involving \({n+r \brace k+r}_{r,Y}\) (see Theorems 2.11, 2.14).
As one of our future projects, we would like to continue to study probabilistic versions of many special polynomials and numbers and to find their applications to physics, science and engineering as well as to mathematics.
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Kim, T., Kim, D.S. Probabilistic Bernoulli and Euler Polynomials. Russ. J. Math. Phys. 31, 94–105 (2024). https://doi.org/10.1134/S106192084010072
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DOI: https://doi.org/10.1134/S106192084010072