As an application of our main results, we study the polynomials satisfying the functional equation
where and fixed integer .
Theorem 3.1 Let a, N be positive integers and such that . Then we have the following properties:
There exists an unique sequence of monic polynomials
Polynomials are eigenfunctions for the operators with eigenvalues , that is
where is the Hurwitz zeta function.
Proof The existence of a sequence of monic polynomials P is satisfied from Theorem 2.1 and Theorem 2.4.
Now we must observe the uniqueness of . For this end, we take two different monic polynomials and of degree n satisfying (2).
Suppose that , where and . From (2) and the definition of , we can write
Subtracting (4) from (3), we get
Identifying the coefficients of on both sides, we have , but this contradicts our stipulations that , , and . Hence, the proof of (i) is completed.
We prove (ii). It is easy to see that
and putting , we obtain
Thanks to Theorem 2.4, we find the generating function of polynomials satisfying (2). More precisely, we have the following theorem.
Theorem 3.2 For all , we have the following results:
The difference formula of
is given by
Proof We prove (i). From (2) and the definition of the operators , we have
we derive this equation and obtain
Since is monic with degree , from Theorem 3.1(i), we arrive at
We prove (ii). For , by taking and , we have
In the above equation, putting and , respectively, we arrive at
Multiplying each side of (7) by and then substrate it from (6), we have the following relation:
Since , we obtain that for all and .
We prove (iii). We can write
On the other hand, by using Theorem 3.2(i), we get
We multiply the each side of (9) by and then substrate (8) from (9), we arrive to
From Theorem 3.2(ii) and equality , we get
Therefore, we obtain the desired result. □
Using Theorem 2.4 and Theorem 3.1, we can establish the following result.
Theorem 3.3 For , the generating function of is given by
Proof Let integer and write
Using the difference formula in Theorem 3.2, we get
We consider the generating function
We compare the coefficients of in the above equation and we obtain
In particular, if we take , then we have
Therefore, we note that the polynomials and are equal on the infinite set . Then we can write for all ,
Now, by derivation on x we get
We obtain the equality
Hence, we obtain the generating function of . □
Remark 3.4 The case of Theorem 3.3 recovers the so-called generalized Bernoulli and Euler polynomials, which are studied in .