Abstract
The aim of this paper is to prove multiplication formulas of the normalized polynomials by using the umbral algebra and umbral calculus methods. Our polynomials are related to the Hermite-type polynomials.
AMS Subject Classification:05A40, 11B83, 11B68.
Similar content being viewed by others
1 Introduction
In this paper, we use the following notations:
Here, we first give some remarks on the normalized polynomials.
Firstly, we introduce some notations which are related to the earlier works by (among others) Carlitz [1, 2], Bodin [3], Roman [[4], pp.1-125]. We recall from the work of Bodin [3]: Let p be a prime number and . For , we denote by the finite field having q elements. denotes the multiplicative group of non-zero elements of .
Let be a polynomial of degree exactly d
f is said to be normalized if the first non-zero term in the sequence is equal to 1. Any polynomial g can be written
where f is a normalized polynomial and (cf. [3]).
We recall the work of Carlitz [[2], p.60]: Let k be a fixed integer >1 and let denote (complex) numbers such that
Let or 0 and let be distinct numbers. Then consider the functional equation
where denotes a normalized polynomial of degree m (that is, a polynomial with the highest coefficient 1). Here is completely determined by (1); moreover, form an Appell set of polynomials (cf. [2]).
Theorem 1.1 Let k be a fixed integer >1 and let be complex numbers such that
Let or 0 and let be distinct numbers. Then equation (1) is satisfied by a unique set of normalized polynomials which form an Appell set (cf. [2]).
Every Appell set satisfies an equation of the form (1) (cf. [2]).
If is a normalized polynomial, then it satisfies the following formula:
If y is an even positive integer, some normalized polynomials satisfy the following equation (cf. [1]):
where and denote the normalized polynomials of degree and n, respectively.
We give some Hermite base polynomials of higher order, which are defined as follows (cf. [5] and [6]):
where denotes Hermite base Bernoulli polynomials of higher order,
where denotes Hermite base Euler polynomials of higher order and
where denotes Hermite base Genocchi polynomials of higher order.
The proof of polynomials which satisfied (2) was given in various ways. In this paper, we study normalized polynomials which are defined above by using the umbral algebra and umbral calculus methods. We also recall from the work of Roman [4] the following.
Let P be the algebra of polynomials in the single variable x over the field complex numbers. Let be the vector space of all linear functionals on P. Let
be the action of a linear functional L on a polynomial . Let denote the algebra of formal power series in the variable t over ℂ. The formal power series
defines a linear functional on P by setting
for all . In a special case,
This kind of algebra is called an umbral algebra (cf. [4]). Any power series
is a linear operator on P defined by
Here, each plays three roles in the umbral calculus: a formal power series, a linear functional and a linear operator. For example, let and
As a linear functional, satisfies the following property:
As a linear operator, satisfies the following property:
Let , be in , then
for all polynomials . The order of a power series is the smallest integer k for which the coefficient of does not vanish. If , . A series for which
is called a delta series. A series for which
is called an invertible series (for details, see [4]).
Theorem 1.2 [[4], p.20, Theorem 2.3.6]
Let be a delta series and let be an invertible series. Then there exists a unique sequence of polynomials satisfying the orthogonality conditions
for all .
The sequence in (6) is the Sheffer polynomials for a pair . The Sheffer polynomials for a pair are the Appell polynomials or Appell sequences for (cf. [4]).
The Appell polynomials are defined by means of the following generating function (cf. [4]):
The Appell polynomials satisfy the following relations:
the derivative formula
and
the multiplication formula
where .
In the next section, we need the following generalized multinomial identity.
Lemma 1.3 (Generalized multinomial identity [[7], p.41, Equation (12m)])
If are commuting elements of a ring (, ), then we have for all real or complex variables α:
the last summation takes places over all positive or zero integers , where
are called generalized multinomial coefficients defined by [[7], p.27, Equation (10c″)], where and .
2 Some identities of normalized polynomials
In this section, we derive some identities and properties related to Hermite base normalized polynomials.
If we set
in (8), we obtain the following lemma.
Lemma 2.1 Let . The following relationship holds true:
If we set
in (8), we obtain the following lemma.
Lemma 2.2 Let . The following relationship holds true:
If we set
in (8), we obtain the following lemma.
Lemma 2.3 Let . The following relationship holds true:
3 Multiplication formulas for normalized polynomials
In this section, we study the Hermite base normalized polynomials. The Hermite base Bernoulli-type polynomials satisfy equation (2), which is given by the following theorem.
Theorem 3.1 Let . The following multiplication formula of the polynomials holds true:
where
Proof By using (12) in (11), we get
Using (13), we have
After some calculations, we obtain
By using Lemma 1.3 in the above equation, we get
where
From (5), we arrive at the desired result. □
The Hermite base Euler-type polynomials and the Hermite base Genocchi-type polynomials satisfy equation (2) for all . But for m being odd, these polynomials in studies normalize condition in (2). For even m, these polynomials satisfy (3).
Theorem 3.2 Let . The following multiplication formula of the polynomials holds true:
when m is odd,
when m is even, where
Proof Let m be odd.
From (11), (14) and (15), we obtain
After some calculations, we get
By using Lemma 1.3, we obtain
where
From (5), we arrive at the desired result.
Let m be even.
From (11), (14) and (15), we obtain
From (13), we get
After some calculations, we have
By using Lemma 1.3, we obtain
where
From (5) and (10), we arrive at the desired result. □
Theorem 3.3 Let . The following multiplication formula of the polynomials holds true:
when m is odd,
when m is even, where
Proof Let m be odd.
From (11), (16) and (17), we obtain
After some calculations, we get
By using Lemma 1.3, we obtain
where
From (5), we arrive at the desired result.
Let m be even.
From (11), (16) and (17), we obtain
Using (13), we get
After some calculations, we have
By using Lemma 1.3, we obtain
where
From (5), we arrive at the desired result. □
Remark 3.4 By substituting and into Theorem 3.1, Theorem 3.2 and Theorem 3.3, one can obtain multiplication formulas for the Bernoulli, Euler and Genocchi polynomials (cf. [1, 2, 4–16]).
Remark 3.5 The proofs of Theorem 3.1, Theorem 3.2 and Theorem 3.3 are also given by the generating functions method and may be other.
References
Carlitz L: A note on the multiplication formulas for the Bernoulli and Euler polynomials. Proc. Am. Math. Soc. 1953, 4: 184-188. 10.1090/S0002-9939-1953-0052569-8
Carlitz L: The multiplication formulas for the Bernoulli and Euler polynomials. Math. Mag. 1953, 27: 59-64. 10.2307/3029762
Bodin A: Number of irreducible polynomials in several variables over finite fields. Am. Math. Mon. 2008, 115: 653-660. arXiv:0706.0157v2 [math.AC]
Roman S: The Umbral Calculus. Dover, New York; 2005.
Dere, R, Simsek, Y: Bernoulli type polynomials on Umbral Algebra. arXiv:1110.1484v1 [math.CA]
Dere R, Simsek Y: Applications of umbral algebra to some special polynomials. Adv. Stud. Contemp. Math. 2012, 22: 433-438.
Comtet L: Advanced Combinatorics: the Art of Finite and Infinite Expansions. Reidel, Dordrecht; 1974. (Translated from the French by J. M. Nienhuys)
Dattoli G, Migliorati M, Srivastava HM: Sheffer polynomials, monomiality principle, algebraic methods and the theory of classical polynomials. Math. Comput. Model. 2007, 45: 1033-1041. 10.1016/j.mcm.2006.08.010
Dere R, Simsek Y: Genocchi polynomials associated with the Umbral algebra. Appl. Math. Comput. 2011, 218: 756-761. 10.1016/j.amc.2011.01.078
Luo Q-M: The multiplication formulas for the Apostol-Bernoulli and Apostol-Euler polynomials of higher order. Integral Transforms Spec. Funct. 2009, 20: 377-391. 10.1080/10652460802564324
Milne-Thomson LM: Two classes of generalized polynomials. Proc. Lond. Math. Soc. 1933, s2-35: 514-522. 10.1112/plms/s2-35.1.514
Simsek, Y: Generating functions for generalized Stirling type numbers, Array type polynomials, Eulerian type polynomials and their applications. http://arxiv.org/pdf/1111.3848v2.pdf
Srivastava, HM, Kurt, B, Simsek, Y: Some families of Genocchi type polynomials and their interpolation functions. Integral Transforms Spec. Funct. (2012), iFirst, 1-20
Srivastava HM: Some generalizations and basic (or q -) extensions of the Bernoulli, Euler and Genocchi polynomials. Appl. Math. Inf. Sci. 2011, 5: 390-444.
Srivastava HM, Kim T, Simsek Y: q -Bernoulli numbers and polynomials associated with multiple q -zeta functions and basic L -series. Russ. J. Math. Phys. 2005, 12: 241-268.
Srivastava HM, Choi J: Zeta and q-Zeta Functions and Associated Series and Integrals. Elsevier, Amsterdam; 2012.
Acknowledgements
Dedicated to Professor Hari M Srivastava.
All authors are partially supported by Research Project Offices Akdeniz Universities. We would like to thank the referees for their valuable comments.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors completed the paper together. All authors read and approved the final manuscript.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Dere, R., Simsek, Y. Normalized polynomials and their multiplication formulas. Adv Differ Equ 2013, 31 (2013). https://doi.org/10.1186/1687-1847-2013-31
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1687-1847-2013-31