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Extended p-adic q-invariant integrals on associated with applications of umbral calculus
Advances in Difference Equations volume 2013, Article number: 96 (2013)
The fundamental aim of this paper is to consider some applications of umbral calculus by utilizing from the extended p-adic q-invariant integral on . From those considerations, we derive some new interesting properties on the extended p-adic q-Bernoulli numbers and polynomials. That is, a systemic study of the class of Sheffer sequences in connection with generating function of the p-adic q-Bernoulli polynomials are given in the present work.
MSC:05A10, 11B65, 11B68, 11B73.
In the complex plane, the Bernoulli polynomials, , are defined by
In particular, the case in (1.1), we have are called Bernoulli numbers. These numbers are extremely important in number theory and other areas of mathematics and physics. With the help of generating function of Bernoulli numbers, one can easily derive that , , , , , , , …, and for (see [1–8]). As is well known, the Riemann zeta function is defined by
We note that the Bernoulli numbers interpolate by the Riemann zeta function, which plays an important role in analytic number theory and has applications in physics, probability theory and applied statistics. Firstly, Leonard Euler studied and introduced the Riemann zeta function in a real argument without using complex analysis. From (1.1) and (1.2), one has
A link between the zeta function and prime numbers was discovered by Euler, who proved the following identity:
Let p be a fixed odd prime number. Throughout this work, we use the following notations, where denotes the ring of p-adic rational integers, ℚ denotes the field of rational numbers, denotes the field of p-adic rational numbers, and denotes the completion of algebraic closure of . Let ℕ be the set of natural numbers and . The p-adic absolute value is defined by . Also, we assume that is an indeterminate. Let be the space of uniformly differentiable functions on . For , Kim defined p-adic q-invariant integral on by the rule:
where is q-analogue of x defined by
Let . By (1.3), we have
In , Kim showed that Carlitz’s q-Bernoulli numbers and polynomials can be expressed as an integral by the q-analogue of the ordinary p-adic invariant measure as follows:
Now also, we consider the extended p-adic q-invariant integral on due to Kim  in the following form: for
where are called extended p-adic q-invariant integral on .
Let us now consider , then we compute as follows:
Therefore, we state the following lemma.
Lemma 1 For ,
Taking in Lemma 1, then we consider the following generating function:
where are called extended q-Bernoulli polynomials. In the special case, , are called extended q-Bernoulli numbers.
We note that
That is, we have
The relation between extended p-adic q-Bernoulli numbers and extended p-adic q-Bernoulli polynomials is given by
with the usual of replacing by . By (1.7) and (1.8), we easily see that
From (1.7), we derive Witt’s formulae for extended p-adic q-Bernoulli numbers and polynomials, respectively:
By (1.7), we have
Let us now consider the following:
By applying Mellin transformation to (1.11), we derive that for :
Here, is Euler’s Gamma function. Thanks to (1.10) and (1.12), we discover the following:
Setting and in (1.13) reduces to
which has a profound effect on number theory and complex analysis.
By (1.6) and (1.7), we develop as follows:
where d is a natural number. That is,
By (1.9) and (1.14), we get
Putting and in (1.15), then it leads to , which is well known as Raabe’s formula.
Let us now define the following notations, where ℂ denotes the set of complex numbers, ℱ denotes the set of all formal power series in the variable t over ℂ with , and denotes the vector space of all linear functional on , denotes the action of the linear functional L on the polynomial , and it is well known that the vector space operation on is defined by and for some constant c in ℂ (see [27–30]).
The following is well known as a formal power series by the rule:
where is the Kronecker delta. It is easy to see that
therefore we procure
and so as linear functionals (see [27–30]). Additionally, the map is a vector space isomorphism from onto ℱ. Henceforth, ℱ will denote both the algebra of the formal power series in t and the vector space of all linear functionals on , and so an element of ℱ will be thought of as both a formal power series and a linear functional. ℱ will be called as umbral algebra (see [27–30]).
Obviously, . From this, it reduces to
and for all polynomial ,
(for details, see [27–30]). The order of the power series is the smallest integer k for which does not vanish. It is considered if . We see that and . The series has a multiplicative inverse, denoted by or , if and only if . Such series is called an invertible series. A series for which is called a delta series (see [27–31]). For , we have . A delta series has a compositional inverse such that .
For , we have . From (1.17), we have
Hence, we get that
By (1.19), we have
So from the above
Let be a polynomial with . Let be a delta series and let be an invertible series. Then there exists a unique sequence of polynomials such that for all . The sequence is called the Sheffer sequence for or that is Sheffer for .
The Sheffer sequence for is called the associated sequence for or is associated to . The Sheffer sequence for is called the Appell sequence for or is Appell for .
Let . Then we have
Let be Sheffer for . Then
Also, it is well known in  that
where and the sum is over all nonnegative integers such that (see ).
Dere and Simsek have studied applications of umbral algebra to special functions in . Kim et al. also gave some properties of umbral calculus for Frobenius-Euler polynomials  and Euler polynomials . Also, they investigated some new applications of umbral calculus associated with p-adic invariants integral on and fermionic p-adic integral on in .
By the same motivation of the above, we also discover both new and interesting applications of umbral calculus by using extended p-adic q-invariant integral on . By virtue of which, we procure some new interesting equalities on the extended p-adic q-Bernoulli numbers and polynomials and extended p-adic q-Bernoulli polynomials of order k. Recently, several authors have studied the q-Bernoulli numbers and polynomials. Also, we note that our q-extensions of Bernoulli numbers and polynomials in the present paper are different from the q-extensions of Bernoulli numbers and polynomials of several authors in previous papers.
2 Identities involving extended p-adic q-invariant integrals on related to applications of umbral calculus
Suppose that is an Appell sequence for . Then, by (1.24), we have
We now consider that
Therefore, we easily notice that is an invertible series. By (2.1), we have
Also, by (1.24), we have
Because of (2.3) and (2.4), we have the following proposition.
Proposition 1 For , is an Appell sequence for .
By (1.9), we have
Because of (2.3) and (2.5), we discover the following:
Therefore, we arrive at the following theorem.
Theorem 1 Let . Then we have for :
By (1.9), it is not difficult to see that
By comparing the coefficients of on the above, we have the following:
By Theorem 1, we derive
So from above
Thus, we have
From (2.7), (2.8) and (2.9), we have the following theorem.
Theorem 2 For , then we have
Suppose that is Sheffer sequence for . Then the following is introduced as Sheffer identity by the rule:
Thanks to (1.7) and (2.11), we have
From the above, we readily see that
By (1.7), we easily get for :
By virtue of (1.15) and (2.12), we see that
Let us now contemplate the linear functional by the following expression:
for all polynomials . From (2.13), we readily derive that
Thus, we have
Therefore, by (2.13) and (2.15), we arrive at the following theorem.
Theorem 3 For , we have
In view of (1.9) and (2.18), we see that
By (1.9) and (2.20), we see that for :
Consequently, we get the following theorem.
Theorem 4 For , we have
For , we introduce extended p-adic q-Bernoulli polynomials of order k as follows:
which we have used the following equality:
In the special case, for in (2.23), we have , which are called extended p-adic q-Bernoulli numbers of order k.
From (2.23), we have
Equating (2.23) and (2.24), we have
From (2.24) and (2.25), we want to note that is a monic polynomial of degree n with coefficients in ℚ. For , let us consider that
From (2.26), we easily see that is an invertible series. On account of (2.23) and (2.26), we derive that
Also, we have
By virtue of (2.27) and (2.28), we easily see that is an Appell sequence for . Then, by (2.27) and (2.28), we get the following theorem.
Theorem 5 For and , we have
In the special case, the extended p-adic q-Bernoulli polynomials of degree k are given by
Thus, we get
Let us take the linear functional that satisfies
for all polynomials . Therefore, we develop as follows:
Therefore, the following theorem can be stated.
Theorem 6 For , we have
From (1.25), we see that
Therefore, we get
Remark 1 Our applications for extended p-adic q-Bernoulli polynomials, extended p-adic q-Bernoulli numbers and extended p-adic q-Bernoulli polynomials of order k seem to be interesting for evaluating at and , which reduce to Bernoulli polynomials and Bernoulli polynomials of order k, are defined respectively by
Also, it is known that these polynomials are expressed by the rule:
where the limits are taken in .
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Dedicated to Professor Hari M Srivastava.
The authors express their sincere thanks to the referee(s) for the careful and detailed reading of the manuscript and very helpful suggestions that improved the manuscript substantially.
The authors declare that they have no competing interests.
All of the authors contributed equally to the manuscript and read and approved the final draft.