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Some identities involving Gegenbauer polynomials
Advances in Difference Equations volume 2012, Article number: 219 (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 .
The Gegenbauer polynomials are given in terms of the Jacobi polynomials with (, ) by
From (1.1), we note that is a polynomial of degree n with real coefficients and . The leading coefficient of is . By the theory of Jacobi polynomials with , , and , we get
It is not difficult to show that is a solution of the following Gegenbauer differential equation:
The Rodrigues formula for the Gegenbauer polynomials is well known as the following:
Equation (1.3) can be easily derived from the properties of Jacobi polynomials.
As is well known, the generating function of Gegenbauer polynomials is given by
Equation (1.4) can be also derived from the generating function of Jacobi polynomials.
From (1.4), we note that
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.
By (1.1) and (1.2), we get
where is the Kronecker symbol and it holds for each fixed with and .
Equation (1.6) implies the orthogonality of and equation (1.6) is important in deriving our results in this paper. From (1.5), we can derive the following derivative of Gehenbauer polynomials :
By (1.7), we get
As is well known, the Bernoulli polynomials are defined by the generating function to be
with the usual convention about replacing by . In the special case, , are called the n th Bernoulli numbers.
From (1.9), we note that
The Euler polynomials are also defined by the generating function to be
with the usual convention about replacing by . In the special case, , are called the n th Euler numbers. By (1.12), we see that the recurrence formula for is given by
For each fixed with and , let be an inner product space with respect to the inner product
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.
2 Some identities involving Gegenbauer polynomials
Let us take , . Then, by (1.6) and (1.14), we get
Thus, from (2.1), we have
By (1.3) and (2.2), we get
Therefore, by (2.3), we obtain the following proposition.
Proposition 2.1 For, let
For example, let . From Proposition 2.1, we note that
Let us assume that (mod2). Then, by (2.4), we get
where is the beta function which is defined by .
It is easy to show that
Therefore, by (2.5) and (2.6), we obtain the following identity:
Let us take . Then, by (1.10), we get
From (1.10) and (2.7), we can derive the following equation:
Let us consider that (mod2). Then, by (2.9), we get
For with (mod2), we have
By (2.10) and (2.11), we get
From (2.8) and (2.12), we have
Therefore, by (2.13) and Proposition 2.1, we obtain the following theorem.
Theorem 2.2 For, we have
By the same method, we get
From (1.1), we note that
Let us take . From Proposition 2.1, can be rewritten as
Then, by Proposition 2.1 and (2.15), we get
It is not difficult to show that
From the fundamental theorem of gamma function, we have
By (2.18) and (2.19), we get
From (2.17) and (2.20), we have
Therefore, by (2.21), we obtain the following theorem.
Theorem 2.3 Forwith, we have
Let us take . Then, from (1.1), we have
In the previous paper, we have shown that
From (2.22) and (2.23), we have
Let . Then, by Proposition 2.1, we get
By (2.24), we get
From (2.26) and (2.27), we have
It is easy to show that
By the fundamental theorem of gamma function, we see that
As is well known, the duplication formula for the gamma function is given by
By (2.29), (2.30), (2.31), and (2.32), we get
From (2.28) and (2.34), we have
Therefore, by (2.35), we obtain the following theorem.
Theorem 2.4 For, we have
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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.
The authors declare that they have no competing interests.
All authors contributed equally to the manuscript and typed, read, and approved the final manuscript.
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Kim, D.S., Kim, T. & Rim, SH. Some identities involving Gegenbauer polynomials. Adv Differ Equ 2012, 219 (2012). https://doi.org/10.1186/1687-1847-2012-219
- Gamma Function
- Product Space
- Information Entropy
- Jacobi Polynomial
- Quantum Mechanical System