- Open Access
Some identities involving Gegenbauer polynomials
© Kim et al.; licensee Springer 2012
- Received: 13 August 2012
- Accepted: 30 November 2012
- Published: 19 December 2012
In this paper, we derive some interesting identities involving Gegenbauer polynomials arising from the orthogonality of Gegenbauer polynomials for the inner product space with respect to the weighted inner product .
- Gamma Function
- Product Space
- Information Entropy
- Jacobi Polynomial
- Quantum Mechanical System
Equation (1.3) can be easily derived from the properties of Jacobi polynomials.
Equation (1.4) can be also derived from the generating function of Jacobi polynomials.
The proof of (1.5) is given in the following book: Stein and Weiss, Introduction to Fourier Analysis in Euclidean Space, Princeton University Press, 1971.
where is the Kronecker symbol and it holds for each fixed with and .
with the usual convention about replacing by . In the special case, , are called the n th Bernoulli numbers.
where . The information entropy of Gegenbauer polynomials is relevant since it is related to the angular part of the information entropies of certain quantum mechanical systems such as the harmonic oscillator and the hydrogen atom in D dimensions. In , Buyarov, Lopez-Artes, Martinez-Finkelshtein, and Van Assche gave an effective method to compute the entropy for Gegenbauer polynomials with an integer parameter and obtain the first few terms in the asymptotic expansion as the degree of the polynomial tends to infinity. That is, an efficient method was provided for evaluating, in a closed form, the information entropy of the Gegenbauer polynomials in the case when . For given values of n and l, this method requires the computation by means of recurrence relations of two auxiliary polynomials, and , of degrees and , respectively (see ). In , Sanchez-Ruiz showed that is related to the coefficients of the Gaussian quadrature formula for the Gegenbauer weights , and this fact is used to obtain the explicit expression of . The position and momentum information entropies of D-dimensional quantum systems with central potentials, such as the isotropic harmonic oscillator and the hydrogen atom, depend on the entropies of the (hyper)spherical harmonics (see ). In turn, these entropies are expressed in terms of the entropies of the Gegenbauer (ultraspherical) polynomials , the parameter λ being either an integer or a half-integer number. Up to now, however, the exact analytical expression of the entropy of Gegenbauer polynomials of arbitrary degree n has only been obtained for the particular values of the parameter (see ). In , de Vicente, Gandy, Sanchez-Ruiz presented a novel approach to the evaluation of the information entropy of Gegenbauer polynomials, which makes use of trigonometric representations for these polynomials and complex integration techniques. Using this method, we are able to find the analytical expression of the entropy for arbitrary values of both n and (see ). The Gegenbauer polynomial seems to be interesting and important in the area of mathematical physics. Recently, many authors have studied Gegenbauer polynomials related to mathematical physics (see [1–5, 7, 10, 11, 14, 18, 21, 22]). In this paper, we derive some interesting identities involving Gegenbauer polynomials arising from the orthogonality of those for the inner product space with respect to the weighted inner product .
Our methods used in this paper are useful in finding some new identities and relations on the Bernoulli and Euler polynomials involving Gegenbauer polynomials.
Therefore, by (2.3), we obtain the following proposition.
where is the beta function which is defined by .
Therefore, by (2.13) and Proposition 2.1, we obtain the following theorem.
Therefore, by (2.21), we obtain the following theorem.
Therefore, by (2.35), we obtain the following theorem.
This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology 2012R1A1A2003786.
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