Abstract
In this paper, we construct some new Christoffel–Darboux type identities for Legendre, Laguerre and Hermite polynomials. We obtain these types of identities for the derivatives of these polynomials.
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Introduction
In [1], we have simplified the fraction
in terms of \(P_i(x)\),\(P_j(y)\) where \(P_n\in \{T_n,U_n,V_n,W_n\}\). Also, for every kind of Chebyshev polynomials, we have obtained the expanded form of the fraction
in terms of \(P_i(x)\), \(P_j(y)\) where \(P_{n}^{(s)}(x)\) is the sth derivative of \(P_{n}(x)\).
In this paper, we expand the fraction (1) where \(P_n(x)\) is Legendre, Laguerre and Hermite polynomials.
Christoffel–Darboux type identities of Legendre, Laguerre and Hermite polynomials
Theorem 2.1
Let \(\{P_n(x)\}_{n=0}^{\infty }\) be a sequence of orthogonal polynomials with respect to the weight function w(x) on interval [a, b] then
where
and \(C_{n+1,k}\) is the coefficient of \(x^k\) in \(P_{n+1}(x)\).
Proof
\(P_n(x)\) is orthogonal to every polynomial of degree less than n. So, if \(i+j>n\) then \( A_{i,j}^{n+1}=0\). If \(i+j\le n \) then use orthogonality and expanded form of \(P_n(x)\) to obtain the result. \(\square \)
Corollary 2.1
If the interval [a, b] is symmetric about the origin and \(P_n(-x)=(-1)^nP_n(x)\) then for even \(n+i+j\), \(A_{i,j}^{n}=0\).
If the linearization formula of \(P_n(x)\) is available then we can compute \(A_{i,j}\) coefficients in Eq. (3) by using one sum instead of using double sum in Eq. (4).
Christoffel–Darboux type identities of Hermite polynomials
Theorem 3.1
Let \(H_n(x)\) be Hermite polynomial of degree n then
where
Proof
First, we prove that
From [5], use the Hilbert transform of \(H_n(y)e^{-y^2}\) to obtain
So
On the other hand, we have
So, if \(m+n\) is even then \(H_{m,n}(x,y)=0\). If \(m+n\) is odd, then use relation (8) and integration by parts and Rodrigue’s formula of Hermite polynomials to obtain
From the relations (7), (9) by using change of the variable \(\pi f=y\), we obtain
So, for odd \(m+n\), we have
The famous linearization formula of Hermite polynomials is [2]
By using the relations (11) and (12), we can obtain \(A_{i,j}^{n}\) in relation (5). \(\square \)
Corollary 3.1
The \(A_{i,j}^{n}\) coefficients in relation (5) can be computed as follows:
Now, we can obtain Christoffel–Darboux type identities for the derivatives of Hermite polynomials.
Corollary 3.2
Let
where
Christoffel–Darboux type identities of Legendre polynomials
Theorem 4.1
Let \(P_n(x)\) be Legendre polynomial of degree n then
where
Proof
Legendre function of the second kind is defined by
and
Therefore
The following famous linearization formula of Legendre polynomials is Neumann-Adams formula [2]:
Now, use the relations (17), (18) and (19) to obtain the result. \(\square \)
Corollary 4.1
The \(A_{i,j}^{n}\) coefficients in relation (15) can be computed as follows:
Now, we can obtain Christoffel–Darboux type identities for the derivatives of Legendre polynomials.
From [4], for the case \(\gamma =0\), we can derive
where
Corollary 4.2
where
Christoffel–Darboux type identities of Laguerre polynomials
The famous linearization formula of associated laguerre polynomials is Feldheim formula [6]
In spite of Hermite and Legendre polynomials, the linearization formula of Laguerre polynomials is presented by double summation. The coefficients \(A_{i,j}^{n} \) of Hermite and Legendre polynomials are obtained from (5) and (16) by one summation, and in the following relations, the \(A_{i,j}^{n} \) coefficients of Laguerre polynomials are given by double summation.
Corollary 5.1
Let \(L_n^m(x)\) be associated laguerre polynomials of degree n then
where
The related formula for Laguerre polynomials of degree n is
where
From [3], we have
Let
then
where
Conclusion
In this paper, we obtained some new Christoffel–Darboux type identities for Legendre, Laguerre and Hermite polynomials. We also obtained these types of identities for the derivatives of these polynomials. These formulas are good theoretically and the correctness of the obtained formulas are checked by Maple 17, and Some of these formulas are not efficient numerically.
References
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Acknowledgments
This work has been funded and supported by Islamic Azad University, Karaj Branch, and the author is thankful to it.
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Arzhang, A. A survey on Christoffel–Darboux type identities of Legendre, Laguerre and Hermite polynomials. Math Sci 9, 193–197 (2015). https://doi.org/10.1007/s40096-015-0167-4
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DOI: https://doi.org/10.1007/s40096-015-0167-4
Keywords
- Christoffel–Darboux identity
- Cauchy kernel
- Legendre polynomials
- Laguerre polynomials
- Hermite polynomials