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 August 01, 2013; Revised September 10, 2013; Accepted September 18, 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.
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