Keywords: Legendre polynomials, Bernoulli polynomials, Euler polynomials, Hermite polynomials, Bernstein polynomials, orthogonality
Turkish Journal of Analysis and Number Theory, 2013 1 (1),
pp 13.
DOI: 10.12691/tjant111
Received November 01, 2013; Revised November 23, 2013; Accepted November 26, 2013
Copyright: © 2013 Science and Education Publishing. All Rights Reserved.
1. Introduction
Legendre polynomials, which are special cases of Legendre functions, are introduced in 1784 by the French mathematician A. M. Legendre (17521833). 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
[1]  S. Araci, D. Erdal and J. J. Seo, A study on the fermionic padic qintegral representation on _{P} associated with weighted qBernstein and qGenocchi polynomials, Abstract and Applied Analysis, Volume 2011 (2011), Article ID 649248, 10 pages. 
 In article  

[2]  A. Bagdasaryan, An elementary and real approach to values of the Riemann zeta function, Phys. Atom. Nucl. 73, 251254, (2010). 
 In article  CrossRef 

[3]  W. N. Bailey, On the product of two Legendre polynomials, Proc. Cambridge Philos. Soc. 29 (1933), 173177. 
 In article  CrossRef 

[4]  B. C. Kellner, On irregular prime power divisors of the Bernoulli numbers, Mathematics of Computation, Volume 76, Number 257, January 2007, Pages 405441. 
 In article  CrossRef 

[5]  T. Kim, Some identities on the qEuler polynomials of higher order and qstirling numbers by the fermionic padic integral on p, Russian J. Math. Phys. 16 (2009), 484491. 
 In article  CrossRef 

[6]  T. Kim, J. Choi, Y. H. Kim and C. S. Ryoo, On qBernstein and qHermite polynomials, Proc. Jangjeon Math. Soc. 14 (2011), no. 2, 215221. 
 In article  

[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 
 In article  

[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. 
 In article  

[9]  L. C. Andrews, Special Functions of Mathematics for Engineerings, SPIE Press, 1992, pages 479. 
 In article  
