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 .
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.
2 Some identities 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.
- Shah M: Some properties associated with generalized integral transform in two variables. Univ. Brasov Lucrari Sti. 1976, 18: 15–27.MathSciNetGoogle Scholar
- de Vicente JI, Gandy S, Sánchez-Ruiz J: Information entropy of Gegenbauer polynomials of integer parameter. J. Phys. A, Math. Theor. 2007, 40: 8345–8361. 10.1088/1751-8113/40/29/010View ArticleMathSciNetMATHGoogle Scholar
- Al-Salam WA, Carlitz L: The Gegenbauer addition theorem. J. Math. Phys. 1963, 42: 147–156.MathSciNetView ArticleMATHGoogle Scholar
- McFadden JA: A diagonal expansion in Gegenbauer polynomials for a class of second-order probability densities. SIAM J. Appl. Math. 1966, 14: 1433–1436. 10.1137/0114111MathSciNetView ArticleMATHGoogle Scholar
- Bayad A, Kim T: Identities involving values of Bernstein, q -Bernoulli, and q -Euler polynomials. Russ. J. Math. Phys. 2011, 18(2):133–143. 10.1134/S1061920811020014MathSciNetView ArticleMATHGoogle Scholar
- Bayad A: Modular properties of elliptic Bernoulli and Euler functions. Adv. Stud. Contemp. Math. 2010, 20(3):389–401.MathSciNetMATHGoogle Scholar
- Buyarov VS, Lopez-Artes P, Martinez-Finkelshtein A, Van Assche W: Information entropy of Gegenbauer polynomials. J. Phys. A, Math. Gen. 2000, 33: 6549–6560. 10.1088/0305-4470/33/37/307MathSciNetView ArticleMATHGoogle Scholar
- Cangul IN, Kurt V, Ozden H, Simsek Y: On the higher-order w - q -Genocchi numbers. Adv. Stud. Contemp. Math. 2009, 19(1):39–57.MathSciNetMATHGoogle Scholar
- Choi J, Kim DS, Kim T, Kim YH: Some arithmetic identities on Bernoulli and Euler numbers arising from the p -adic integrals on . Adv. Stud. Contemp. Math. 2012, 22(2):239–247.MathSciNetMATHGoogle Scholar
- Khan S, Al-Gonah AA, Yasmin G: Generalized and mixed type Gegenbauer polynomials. J. Math. Anal. Appl. 2012, 390(1):197–207. 10.1016/j.jmaa.2012.01.026MathSciNetView ArticleMATHGoogle Scholar
- Kim T, Choi J, Kim YH, Ryoo CS: On q -Bernstein and q -Hermite polynomials. Proc. Jangjeon Math. Soc. 2011, 14(2):215–221.MathSciNetMATHGoogle Scholar
- Kim T: Some identities on the q -Euler polynomials of higher order and q -Stirling numbers by the fermionic p -adic integral on . Russ. J. Math. Phys. 2009, 16(4):484–491. 10.1134/S1061920809040037MathSciNetView ArticleMATHGoogle Scholar
- Kim T: Symmetry of power sum polynomials and multivariate fermionic p -adic invariant integral on . Russ. J. Math. Phys. 2009, 16(1):93–96. 10.1134/S1061920809010063MathSciNetView ArticleMATHGoogle Scholar
- Kim DS, Kim T, Dolgy DV: Some identities on Bernoulli and Hermite polynomials associated with Jacobi polynomials. Discrete Dyn. Nat. Soc. 2012., 2012: Article ID 584643Google Scholar
- Ozden H, Cangul IN, Simsek Y: Remarks on q -Bernoulli numbers associated with Daehee numbers. Adv. Stud. Contemp. Math. 2009, 18(1):41–48.MathSciNetMATHGoogle Scholar
- Rim S-H, Lee S-J: Some identities on the twisted -Genocchi numbers and polynomials associated with q -Bernstein polynomials. Int. J. Math. Math. Sci. 2011., 2011: Article ID 482840Google Scholar
- Ryoo CS: Some relations between twisted q -Euler numbers and Bernstein polynomials. Adv. Stud. Contemp. Math. 2011, 21(2):217–223.MathSciNetMATHGoogle Scholar
- Sanchez-Ruiz J: Information entropy of Gegenbauer polynomials and Gaussian quadrature. J. Phys. A, Math. Gen. 2003, 36: 4857–4865. 10.1088/0305-4470/36/17/312MathSciNetView ArticleMATHGoogle Scholar
- Simsek Y: Special functions related to Dedekind-type DC-sums and their applications. Russ. J. Math. Phys. 2010, 17(4):495–508. 10.1134/S1061920810040114MathSciNetView ArticleMATHGoogle Scholar
- Simsek Y: Generating functions of the twisted Bernoulli numbers and polynomials associated with their interpolation functions. Adv. Stud. Contemp. Math. 2008, 16(2):251–278.MathSciNetMATHGoogle Scholar
- Sneddon IN: The evaluation of an integral involving the product of two Gegenbauer polynomials. SIAM Rev. 1967, 9: 569–572. 10.1137/1009078MathSciNetView ArticleMATHGoogle Scholar
- Shah M: Applications of Gegenbauer (ultraspherical) polynomials in cooling of a heated cylinder. An. Univ. Timisoara, Ser. Sti. Mat. 1970, 8: 207–212.MathSciNetMATHGoogle Scholar
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