Probabilistic degenerate Fubini polynomials associated with random variables

Let Y be a random variable such that the moment generating function of Y exists in a neighborhood of the origin. The aim of this paper is to study probabilistic versions of the degenerate Fubini polynomials and the degenerate Fubini polynomials of order $r$, namely the probabilisitc degenerate Fubini polynomials associated with Y and the probabilistic degenerate Fubini polynomials of order r associated with Y. We derive some properties, explicit expressions, certain identities and recurrence relations for those polynomials.

Let Y be a random variable satisfying the moment condition (see (17)).The aim of this paper is to study probabilistic versions of the degenerate Fubini polynomials and the degenerate Fubini polynomials of order r, namely the probabilisitc degenerate Fubini polynomials associated with Y and the probabilistic degenerate Fubini polynomials of order r associated with Y .We derive some properties, explicit expressions, certain identities and recurrence relations for those polynomials and numbers.In addition, we consider the special cases that Y is the gamma random variable with parameters α, β > 0, the Poisson random variable with parameter α(> 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 degenerate exponentials, the degenerate Stirling numbers of the second kind n k λ , the degenerate Bell polynomials, the degenerate Fubini polynomials and the degenerate Fubini polynomials of order r.We remind the reader of Lah numbers and the partial Bell polynomials.Assume that Y is a random variable such that the moment generating function of Y , E[e tY ] = ∞ n=0 t n n! E[Y n ], (|t| < r), exists for some r > 0. Let (Y j ) j≥1 be a sequence of mutually independent copies of the random variable Y , and let , with S 0 = 0. Then we recall the probabilistic degenerate Stirling numbers of the second kind associated with Y and the probabilistic degenerate Bell polynomials associated with Y , φ Y n,λ (x).Also, we remind the reader of the gamma random variable with parameters α, β > 0. Section 2 is the main result of this paper.Let (Y j ) j≥1 , S k , (k = 0, 1, . . . ) be as in the above.Then we first define the probabilistic degenerate Fubini polynomials associated with the random variable Y , F Y n,λ (x).We derive for F Y n,λ (x) an explicit expression in Theorem 2.1 and an expression as an infinite sum involving E[(S k ) n,λ ] in Theorem 2.2.In Theorem 2.3, when Y ∼ Γ(1, 1), we find an expression for F Y n,λ (x) in terms of Lah numbers and Stirling numbers of the first kind.We obtain a representation of F Y n,λ (x) as an integral over (0, ∞) of the integrand involving φ Y n,λ (x) and its generalization in Theorem 2.14.In Theorem 2.5, we express the probabilistic degenerate Fubini numbers associated with , as a finite sum involving the partial Bell polynomials.Then we introduce the probabilistic degenerate Fubini polynomials of order r associated with Y and deduce an explicit expression for them in Theorem 2.6.We obtain a recurrence relation for F Y n,λ (x) in Theorem 2.7, and another one in Theorem 2.8 together with its generalization in Theorem 2.15.In Theorem 2.9, the rth derivative of F Y n,λ (x) is expressed in terms of F (r+1,Y ) i,λ (x).We get the identity x i in Theorem 2.10 and its generalization in Theorem 2.13.In Theorem 2.11, when Y is the Poisson random variable with parameter α, we express F Y n,λ (x) in terms of the Fubini polynomials F i (x) and n i λ .In Theorem 12, when Y is the Poisson random variable with parameter α, we show is the Bernoulli random variable with probability of success p.For the rest of this section, we recall the facts that are needed throughout this paper.
In [13], the degenerate Stirling numbers of the second kind are defined by It is well known that the degenerate Bell polynomials are defined by (7) e x(e λ (t)−1) = ∞ n=0 φ n,λ (x) t n n! , (see [12 − 14]).
For any integer k ≥ 0, the partial Bell polynomials are given by ( 15) where ( 16) Let Y be a random variable such that the moment generating function of Y Assume that (Y j ) j≥1 is a sequence of mutually independent copies of Y and The probabilistic degenerate Stirling numbers of the second kind associated with random variable Y are defined by [15]).
We recall that Y is the gamma random variable with parameter α, β > 0 if probability density function of Y is given by which is denoted by Y ∼ Γ(α, β).

Probabilistic degenerate Fubini polynomials associated with random variables
Let (Y k ) k≥1 be a sequence of mutually independent copies of random variable Y , and let Now, we consider the probabilistic degenerate Fubini polynomials associated with random variable Y which are given by ( 23) are called the probabilistic degenerate Fubini numbers associated with random variable Y .
From (23), we note that Therefore, by comparing the coefficients on both sides of (24), we obtain the following theorem.
Theorem 2. For n ≥ 0, we have In particular, for Y = 1, we have where S 1 (n, l) are the Stirling numbers of the first kind.Here we should observe that, for all t with |t| small, we have x | is bounded on (0, ∞).Therefore, by comparing the coefficients on both sides of (26), we obtain the following theorem.
Theorem 6.For n ≥ 0, we have By (23), we get Therefore, by comparing the coefficients on both sides of (34), we obtain the following theorem.
Theorem 7.For n ≥ 1, we have From (23), we note that Therefore, by comparing the coefficients on both sides of ( 26), we obtain the following theorem.
Theorem 8.For n ≥ 0, we have Therefore, by (36), we obtain the following theorem.
Theorem 9.For r, n ≥ 0, we have From (19), we note that Therefore, by comparing the coefficients on both sides of (37), we obtain the following theorem.
From (38), we have where F i (x) are the Fubini polynomials given by Therefore, by (39), we obtain the following theorem.
Theorem 11.Let Y be the Poisson random variable with parameter α(> 0).Then we have Let Y be the Poisson random variable with parameter α > 0. Then we have Thus, by ( 40) and (41), we get From Theorem 2.10 and (42), we have Therefore, by (43), we obtain the following theorem.
Theorem 12. Let Y be the Poisson random variable with parameter α(> 0).For n ≥ 0, we have By using Theorem 2.6 and ( 19), we note that Therefore, by (44), we obtain the following theorem.
Theorem 13.For n ≥ 0, we have When Y = 1, we have Now, we observe from ( 21) and Theorem 2.6 that Therefore, by (45), we obtain the following theorem.
From (32) and Theorem 2.14, we have By (46), we get where r is a positive integer.
We observe that Therefore, by (47), we obtain the following theorem.
Theorem 15.For n ≥ 0, we have Let Y be the Bernoulli random variable with probability of success p. Then we have Therefore, by comparing the coefficients on both sides of (49), we obtain the following theorem.
Theorem 16.Let Y be the Bernoulli random variable with probability of success p.
For n ≥ 0, we have .

Conclusion
In this paper, we studied by using generating functions the probabilistic degenerate Fubini polynomials associated with Y and the probabilistic degenerate Fubini polynomials of order r associated with Y , as probabilistic versions of the degenerate Fubini As one of our future projects, we would like to continue to study degenerate versions, λ-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.

Ethical
Statement The submitted article is an original research paper and has not been published anywhere else.There are no clinical trials, animal research, or human trials included in this paper.Funding Tis research was funded by the National Natural Science Foundation of China (No.12271320) and Key Research and Development Program of Shaanxi (No. 2023 ZDLGY-02).
polynomials and the degenerate Fubini polynomials of order r, respectively.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 derived several explicit expressions ofF Y n,λ(x) (see Theorems 2.1, 2.2, 2.4) and those of F r,Y n,λ (x) (see Theorems 2.6, 2.14).We obtained a recurrence relations for F Y n,λ (x) (see Theorem 2.7), and another one (see Theorem 2.8) together with its generalization (see Theorem 2.15).We expressed the rth derivative of F Y n,λ (x) in terms of F = ∞ i=0 E[(S i ) n,λ ]x i (see Theorem 2.10) and its generalization (see Theorem 2.13).We deduced an explicit expression for F Y n,λ (x) when Y ∼ Γ(1, 1) (see Theorem 2.3) and also that when Y is the Poisson random variable with parameter α (see Theorem 2.11).We proved that 1 1−x F Y = ∞ k=0 φ n,λ (kα)x k when Y is the Poisson random variable with parameter α (see Theorem 2.12).We showed F Y n,λ (x) = F n,λ (xp) when Y be the Bernoulli random variable with probability of success p (see Theorem 2.16).