The Legendre Polynomials Associated with Bernoulli, Euler, Hermite and Bernstein Polynomials

Serkan Araci, Mehmet Acikgoz, Armen Bagdasaryan, Erdoğan Şen

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The Legendre Polynomials Associated with Bernoulli, Euler, Hermite and Bernstein Polynomials

Serkan Araci1,, Mehmet Acikgoz2, Armen Bagdasaryan3, Erdoğan Şen4

1Atatürk Street, 31290 Hatay, Turkey

2University of Gaziantep, Faculty of Science and Arts, Department of Mathematics, Gaziantep, Turkey

3Russian Academy of Sciences, Institute for Control Sciences, Profsoyuznaya, Moscow, Russia

4Department of Mathematics, Faculty of Science and Letters, Namik Kemal University, Tekirdağ, Turkey

Abstract

In the present paper, we deal mainly with arithmetic properties of Legendre polynomials by using their orthogonality property. We show that Legendre polynomials are proportional with Bernoulli, Euler, Hermite and Bernstein polynomials.

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Cite this article:

  • Araci, Serkan, et al. "The Legendre Polynomials Associated with Bernoulli, Euler, Hermite and Bernstein Polynomials." Turkish Journal of Analysis and Number Theory 1.1 (2013): 1-3.
  • Araci, S. , Acikgoz, M. , Bagdasaryan, A. , & Şen, E. (2013). The Legendre Polynomials Associated with Bernoulli, Euler, Hermite and Bernstein Polynomials. Turkish Journal of Analysis and Number Theory, 1(1), 1-3.
  • Araci, Serkan, Mehmet Acikgoz, Armen Bagdasaryan, and Erdoğan Şen. "The Legendre Polynomials Associated with Bernoulli, Euler, Hermite and Bernstein Polynomials." Turkish Journal of Analysis and Number Theory 1, no. 1 (2013): 1-3.

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1. Introduction

Legendre polynomials, which are special cases of Legendre functions, are introduced in 1784 by the French mathematician A. M. Legendre (1752-1833). Legendre functions are a vital and important in problems including spherical coordinates. Due to their orthogonality properties they are also useful in numerical analysis (see [9]). Besides, the Legendre polynomials, , are described via the following generating function:

(1)

Legendre polynomials are the everywhere regular solutions of Legendre’s differential equation that we can write as follows:

where and . Taking in (1) and by using geometric series, we see that so that the Legendre polynomials are normalized.

Legendre polynomials can be generated using Rodrigue’s formula as follows:

(2)

Note that the right hand side of (2) is a polynomial (see [3, 9]).

The Bernoulli polynomials are defined by means of the following generating function:

(3)

By (3), we know that Taking in (3), we have that stands for Bernoulli number.

The Euler polynomials are known to be defined as:

(4)

The Euler polynomials can also be expressed by explicit formulas, e.g.

where means the Euler numbers. These numbers are expressed with the Euler polynomials through

Now also, we give the definition of Hermite polynomials as follows:

(5)

Let be the space of continuous functions on For Bernstein operator for is defined by

where and is the set of natural numbers. Here is called Bernstein polynomials, which are defined by

(6)

In [9], [3], the orthogonality of Legendre polynomials is known as

(7)

where is Kronecker’s delta.

In [7], by using orthogonality property of Legendre, Kim et al. effected interesting identities for them. We also obtain some interesting properties of the Legendre polynomials arising from Bernoulli, Euler, Hermite and Bernstein polynomials.

2. Identities on the Legendre Polynomials Arising from Bernoulli, Euler, Hermite and Bernstein Polynomials

Let Then we define an inner product on as follows:

(8)

Note that are the orthogonal basis for Let us now consider then we see that

(9)

where the coefficients are defined over the field of real numbers.

From the above, we readily see that

(10)

By (9) and (10), we have the following proposition.

Proposition 2.1. Let and then

If we take in Proposition (2.1), the coefficients can be found as

(11)

Let Then by using Proposition 2.1 and (11), we have

where are the aforementioned Bernoulli polynomials that can be expressed through Bernoulli numbers as follows:

From this, we have

Therefore we have the following theorem.

Theorem 2.2. Let Then we have

Let By Proposition 2.1 and (11), we have the following theorem.

Theorem 2.3. Let Then we have

Let the Bernstein polynomials By Proposition 2.1 and (11), we have following theorem.

Theorem 2.4. Let We have

The following equality is defined by Kim et al. in [7]:

(12)

Let By Proposition 2.1 and (11), we get the following theorem.

Theorem 2.5. Let Then we have

Let In [8], Kim et al. derived convolution formula for the Euler polynomials as

By Proposition 2.1 and (11), we get the following theorem.

Theorem 2.6. The following equality holds true:

Remark 2.7. By using Theorem 2.1, we can find many interesting identities for the special polynomials in connection with Legendre polynomials.

References

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[2]  A. Bagdasaryan, An elementary and real approach to values of the Riemann zeta function, Phys. Atom. Nucl. 73, 251-254, (2010).
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[3]  W. N. Bailey, On the product of two Legendre polynomials, Proc. Cambridge Philos. Soc. 29 (1933), 173-177.
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[4]  B. C. Kellner, On irregular prime power divisors of the Bernoulli numbers, Mathematics of Computation, Volume 76, Number 257, January 2007, Pages 405-441.
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[5]  T. Kim, Some identities on the q-Euler polynomials of higher order and q-stirling numbers by the fermionic p-adic integral on p, Russian J. Math. Phys. 16 (2009), 484-491.
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[6]  T. Kim, J. Choi, Y. H. Kim and C. S. Ryoo, On q-Bernstein and q-Hermite polynomials, Proc. Jangjeon Math. Soc. 14 (2011), no. 2, 215-221.
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[7]  D. S. Kim, S.-H. Rim and T. Kim, Some identities on Bernoulli and Euler polynomials arising from orthogonality of Legendre polynomials, Journal of Inequalities and Applications 2012, 2012:227
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[8]  D. S. Kim, T. Kim, S.-H. Lee, Y.-H. Kim, Some identities for the product of two Bernoulli and Euler polynomials. Adv. Diff. Equ. 2012; 2012:95.
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[9]  L. C. Andrews, Special Functions of Mathematics for Engineerings, SPIE Press, 1992, pages 479.
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