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ABSTRACT
Recently, many authors have studied degenerate Bernoulli and degenerate Euler polynomials. Let be a random variable whose moment generating function exists in a neighbourhood of the origin. The aim of this paper is to introduce and study the probabilistic extension of degenerate Bernoulli and degenerate Euler polynomials, namely the probabilistic degenerate Bernoulli polynomials associated with
and the probabilistic degenerate Euler polynomials associated with
. Also, we intoduce the probabilistic degenerate
-Stirling numbers of the second associated with
and the probabilistic degenerate two variable Fubini polynomials associated with
. We obtain some properties, explicit expressions, recurrence relations and certain identities for those polynomials and numbers. As special cases of
, we treat the gamma random variable with parameters
, the Poisson random variable with parameter
, and the Bernoulli random variable with probability of success
.
1. Introduction
In Citation[1], Carlitz initiated a study of degenerate versions of Bernoulli and Euler polynomials, namely the degenerate Bernoulli and degenerate Euler polynomials. In recent years, a lot of work has been done for various degenerate versions of many special polynomials and numbers. For example, we found the degenerate Stirling numbers of the first kind and the second kind which turned out to be very important in studying degenerate versions of special polynomials and numbers. It is also remarkable that degenerate umbral calculus and degenerate gamma function were developed along the way.
Let be a random variable satisfying the moment condition (see (13)). The aim of this paper is to study, as probabilistic extensions of degenerate Bernoulli and degenerate Euler polynomials, the probabilistic degenerate Bernoulli polynomials associated with
and the probabilistic degenerate Euler polynomials associated with
, along with the probabilistic degenerate
-Stirling numbers of the second kind associated with
and the probabilistic degenerate two variable Fubini polynomials associated with
.
We derive some properties, explicit expressions, certain identities and recurrence relations for those polynomials and numbers. In addition, as special cases of , we consider the gamma random variable with parameters
, the Poisson random variable with parameter
, and the Bernoulli random variable with probability of success
.
The outline of this paper is as follows. In Section 1, we recall the degenerate exponentials, the degenerate Bernoulli polynomials and the degenerate Euler polynomials. We remind the reader of the degenerate Stirling numbers of the first and the second kinds, and the degenerate -Stirling numbers of the second kind. We recall the degenerate Fubini polynomials and the degenerate two variable Fubini polynomials. Assume that
is a random variable such that the moment generating function of
,
, exists for some
. Let
be a sequence of mutually independent copies of the random variable
, and let
, with
. Then we recall the probabilistic degenerate Stirling numbers of the second kind associated with
,
and the probabilistic degenerate two variable Fubini polynomials associated with
,
. Section 2 includes the main results of this paper. Let
be as in the above. We define the probabilistic degenerate Bernoulli polynomials associated with
,
(see (21)). Then we find explicit expressions for those polynomials in Theorems 1, 2 and 6. We get respectively in Theorems 3, 4 and 5 probabilistic degenerate analogues, involving
and
, of the well-known identities for Bernoulli numbers and polynomials, namely
,
, and
. Here
is the Kronecker’s delta so that it is 1 if
and 0 otherwise. We determine
when
is the gamma random variable with parameters
(see (19)) in Theorem 8 and the Bernoulli random variable with probability of success
in Theorem 9. In Theorem 7, we obtain an identity involving
and
. Then we define the probabilistic degenerate
-Stirling numbers of the second kind associated with
,
and obtain an expression for them in Theorem 10. In Theorem 13, we derive a generalization of the identity in Theorem 7 which involves
and
. We deduce an explicit expression for
in Theorem 11 and that for
, (
with
), in Theorem 12. We define the probabilistic degenerate Euler polynomials associated with
,
(see (45)). We find an explicit expression for
in Theorem 14 and that for
in Theorem 15. In Theorem 16, we obtain a probabilistic degenerate analogue, involving
and
, of the well-known identity for Euler numbers and polynomials, namely
, for any even positive integer
. We derive an explicit expression for
when
is the gamma random variable with parameters
in Theorem 17 and that for
when
is the Poisson random variable with parameter
in Theorem 18. For the rest of this section, we recall the facts that are needed throughout this paper.
For any nonzero , the degenerate exponentials are defined by
where
Carlitz considered the degenerate Bernoulli polynomials defined by
When ,
are called the degenerate Bernoulli numbers. It is immediate to see from (2) that
Note that , where
are the ordinary Bernoulli polynomials given by
The degenerate Euler polynomials are defined as
When ,
are called the degenerate Euler numbers. The values of
can be determined from the recurrence relation (see (Kim et al. Citation23)):
We readily see from (4) that
Note that , where
are the ordinary Euler polynomials given by
It is well known that the Stirling numbers of the first kind are defined as
where
As the inversion formula of (7), the Stirling numbers of the second kind are given by
The degenerate Stirling numbers of the second kind are defined by
Note that . The values of
can be determined from the following recurrence relations (see (Kim and Kim Citation14)):
Also, the degenerate Stirling numbers of the first kind are defined by
From (9), we can easily see that
where is Kronecker's symbol.
Let be a nonnegative integer. Then the degenerate
-Stirling numbers of the second kind are defined by
From (12), we note that
where is a nonnegative integer.
The degenerate Fubini polynomials are given by
Thus, by (14), we get
The degenerate two variable Fubini polynomials are defined by
Note that .
Assume that is a random variable such that the moment generating function of
where stands for the mathematical expectation.
Let be a sequence of mutually independent copies the random variable
, and let
with
.
The probabilistic degenerate Stirling numbers of the second kind associated with are defined by
Note that if
.
Recently, the probabilistic degenerate two variable Fubini polynomials associated with are given by
When ,
are called the probabilistic degenerate Fubini polynomials associated with
.
2. Probabilistic degenerate Bernoulli and degenerate Euler polynomials
A continuous random variable whose density function is given by
for some is said to be the gamma random variable with parameters
, which is denoted by
, (see (Leon-Garcia Citation26; Simsek Citation33)).
Let be a sequence of mutually independent copies of random variable
, and let
We define the probabilistic degenerate Bernoulli polynomials associated with by
When ,
.
For ,
are called the probabilistic degenerate Bernoulli numbers associated with
.
From (21), we note that
Thus, by (22), we get
Thus, by comparing the coefficients on both sides of (23), we obtain the following theorem.
Theorem 1
For , we have
By binomial expansion, we have
Thus, by (21) and (24), we get
Therefore, by comparing the coefficients on both sides of (25), we obtain the following theorem.
Theorem 2.
For , we have
By (21), we get
On the other hand, by (20), we get
Therefore, by (26) and (27), we obtain the following theorem.
Theorem 3.
For , we have
From (21), we have
By comparing the coefficients on both sides of (28), we obtain the following theorem.
Theorem 4.
Let be a nonnegative integer. Then we have
Let be a positive integer. Then we have
Therefore, by (21) and (29), we obtain the following theorem.
Theorem 5
Let be a positive integer. For
, we have
Let be the Poisson random variable with parameter
. We denote this random variable by
. Then we have
By comparing the coefficients on both sides of (30), we obtain the following theorem.
Theorem 6.
Let be the Poisson random variable with parameter
. Then we have
where is a nonnegative integer.
From (15), we note that
Thus, by (31), we get
Therefore, by (31) and (32), we obtain the following theorem.
Theorem 7.
For , we have
In particular, for , we have
Let . Then we get
From (21) and (33), we note that
where are the Cauchy numbers given by
Therefore, by (34), we obtain the following theorem.
Theorem 8.
For , we have
Let be the Bernoulli random variable with probability success
. Then we have
From (21) and (36), we note that
Therefore, by (37), we obtain the following theorem.
Theorem 9
Let be the Bernoulli random variable with probability success
. Then we have
Now, we define the probabilistic degenerate -Stirling numbers of the second kind associated with
as
where is a nonnegative integer.
When , we have
.
From (38), we note that
Therefore, by (39), we obtain the following theorem.
Theorem 10.
For , we have
From (18), we have
Therefore, by (40), we obtain the following theorem.
Theorem 11.
For , we have
Let be a nonnegative integer. Then we have
Therefore, by (41), we obtain the following theorem.
Theorem 12.
Let be a nonnegative integer. Then we have
From (42), we note that
By (18), we get
Therefore, by (43) and (44), we obtain the following theorem.
Theorem 13.
For , we have
We define the probabilistic degenerate Euler polynomials associated with by
When ,
. For
,
are called the probabilistic Euler numbers associated with
.
From (45), we note that
Therefore, by (46), we obtain the following theorem.
Theorem 14.
For , we have
From (45), we note that
Thus, by (47), we get the next result.
Theorem 15.
For , we have
For with
, we have
On the other hand, by (20), we get
Therefore, by (48) and (49), we obtain the following theorem.
Theorem 16.
For with
, we have
Let . Then we have
Thus, by (45) and (50), we get
Therefore, by (51), we obtain the following theorem.
Theorem 17.
For , we have
Let be the Poisson random variable with parameter
. Then we have
Therefore, by (52), we obtain the following theorem.
Theorem 18.
Let be the Poisson random variable with parameter
. Then we have
3. Illustrations of ![](//:0)
for ![](//:0)
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We illustrate our results in Theorem 18 for . By using (5), we get the following values of
, for
:
Then, by using (5), we compute , for
, as in the following:
Finally, from Theorem 18, (54) and , we obtain , for
, when
is the Poisson random variable with parameter
(see ):
Table 1. Values of .
4. Conclusion
Let be a random variable such that the moment generating function of
exists in a neighbourhood of the origin. In this paper, we studied by using generating functions probabilistic extensions of several special polynomials, namely the probabilistic degenerate Bernoulli polynomials associated with
and the probabilistic degenerate Euler polynomials associated with
, together with the probabilistic degenerate
-Stirling numbers of the second associated with
and the probabilistic degenerate two variable Fubini polynomials associated with
. In more detail, we obtained several explicit expressions for
(see Theorems 1, 2, 6) and an explicit expression for each of
, and
(see Theorems 11, 12, 14). We derived three identities about probabilistic degenerate extensions of well-known identities on Bernoulli numbers and polynomials (see Theorems 3-5). Further, we obtained one identity about probabilistic degenerate extensions of well-known identity on Euler numbers and polynomials (see Theorem 16). We obtained an identity involving
and
in Theorem 7 and a generalization of that identity involving
and
in Theorem 13. We determined explicit expressions for
when
in Theorem 8 and
is the Bernoulli random variable with probability of success
in Theorem 9. We found explicit expressions for
when
in Theorem 17 and
is the Poisson random variable with parameter
in Theorem 18. We deduced an explicit expression for
in Theorem 10.
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
Disclosure statement
No potential conflict of interest was reported by the author(s).
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References
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