- Open Access
Some identities on Bernoulli and Euler polynomials arising from orthogonality of Legendre polynomials
© Kim et al.; licensee Springer 2012
- Received: 26 June 2012
- Accepted: 25 September 2012
- Published: 9 October 2012
The purpose of this paper is to investigate some interesting identities on the Bernoulli and Euler polynomials arising from the orthogonality of Legendre polynomials in the inner product space .
- Differential Equation
- Generate Function
- Gamma Function
- Product Space
- Legendre Polynomial
It is a polynomial of degree n. If n is even or odd, then is accordingly even or odd. They are determined up to constant and normalized so that .
where is the Kronecker symbol.
with the usual convention about replacing by .
In the special case, , are called the Bernoulli numbers.
with the usual convention about replacing by .
From (1.2), we can show that is an orthogonal basis for . In this paper, we derive some interesting identities on the Bernoulli and Euler polynomials from the orthogonality of Legendre polynomials in .
Therefore, by (2.1) and (2.3), we obtain the following proposition.
Let us assume that .
where () is the gamma function.
Therefore, by (2.6) and (2.7), we obtain the following Proposition 2.2.
Therefore, by Proposition 2.1, (2.9) and (2.10), we obtain the following theorem.
Therefore, by Proposition 2.1 and (2.17), we obtain the following theorem.
where is the Hermite polynomial of degree n (see ).
The authors would like to express their sincere gratitude to referee for his/her valuable comments and information. This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science, and Technology 2012R1A1A2003786.
- Carlitz L: Some integrals containing products of Legendre polynomials. Arch. Math. 1961, 12: 334–340. 10.1007/BF01650571MathSciNetView ArticleMATHGoogle Scholar
- Carlitz L: Some congruence properties of the Legendre polynomials. Math. Mag. 1960/1961, 34: 387–390.MathSciNetView ArticleGoogle Scholar
- Carlitz L: Some arithmetic properties of the Legendre polynomials. Acta Arith. 1958, 4: 99–107.MathSciNetMATHGoogle Scholar
- Al-Salam WA, Carlitz L: Finite summation formulas and congruences for Legendre and Jacobi polynomials. Monatshefte Math. 1958, 62: 108–118. 10.1007/BF01301283MathSciNetView ArticleMATHGoogle Scholar
- Carlitz L: Some arithmetic properties of the Legendre polynomials. Proc. Camb. Philos. Soc. 1957, 53: 265–268. 10.1017/S0305004100032278View ArticleMathSciNetMATHGoogle Scholar
- Carlitz L: Note on Legendre polynomials. Bull. Calcutta Math. Soc. 1954, 46: 93–95.MathSciNetMATHGoogle Scholar
- Carlitz L: Congruence properties of the polynomials of Hermite, Laguerre and Legendre. Math. Z. 1954, 59: 474–483.MathSciNetView ArticleMATHGoogle Scholar
- Ding D, Yang J: Some identities related to the Apostol-Euler and Apostol-Bernoulli polynomials. Adv. Stud. Contemp. Math. 2010, 20(1):7–21.MathSciNetMATHGoogle Scholar
- Kim G, Kim B, Choi J: The DC algorithm for computing sums of powers of consecutive integers and Bernoulli numbers. Adv. Stud. Contemp. Math. 2008, 17(2):137–145.MathSciNetMATHGoogle Scholar
- Kim T: A note on q -Bernstein polynomials. Russ. J. Math. Phys. 2011, 18(1):73–82. 10.1134/S1061920811010080MathSciNetView ArticleMATHGoogle 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 T: q -Bernoulli numbers and polynomials associated with Gaussian binomial coefficients. Russ. J. Math. Phys. 2008, 15(1):51–57.MathSciNetView ArticleMATHGoogle Scholar
- Kim, T, Kim, DS, Dolgy, DV, Rim, SH: Some identities on the Euler numbers arising from Euler basis polynomials. ARS Comb. 109 (2013, in press)Google Scholar
- Kudo A: A congruence of generalized Bernoulli number for the character of the first kind. Adv. Stud. Contemp. Math. 2000, 2: 1–8.MathSciNetMATHGoogle Scholar
- Leyendekkers JV, Shannon AG, Wong GCK: Integer structure analysis of the product of adjacent integers and Euler’s extension of Fermat’s last theorem. Adv. Stud. Contemp. Math. 2008, 17(2):221–229.MathSciNetMATHGoogle 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 SJ: 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
- Rim SH, Jin JH, Moon EJ, Lee SJ: Some identities on the q -Genocchi polynomials of higher-order and q -Stirling numbers by the fermionic p -adic integral on . Int. J. Math. Math. Sci. 2010., 2010: Article ID 860280Google Scholar
- Ryoo CS: A note on the Frobenius-Euler polynomials. Proc. Jangjeon Math. Soc. 2011, 14(4):495–501.MathSciNetMATHGoogle Scholar
- Ryoo CS: Some identities of the twisted q -Euler numbers and polynomials associated with q -Bernstein polynomials. Proc. Jangjeon Math. Soc. 2011, 14(2):239–248.MathSciNetMATHGoogle Scholar
- Ryoo CS: Some relations between twisted q -Euler numbers and Bernstein polynomials. Adv. Stud. Contemp. Math. 2011, 21(2):217–223.MathSciNetMATHGoogle Scholar
- Simsek Y, Acikgoz M:A new generating function of Bernstein-type polynomials and their interpolation function. Abstr. Appl. Anal. 2010., 2010: Article ID 769095Google 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
- Simsek Y:Complete sum of products of -extension of Euler polynomials and numbers. J. Differ. Equ. Appl. 2010, 16(11):1331–1348. 10.1080/10236190902813967MathSciNetView ArticleMATHGoogle Scholar
- Zhang Z, Yang H: Some closed formulas for generalized Bernoulli-Euler numbers and polynomials. Proc. Jangjeon Math. Soc. 2008, 11(2):191–198.MathSciNetMATHGoogle Scholar
- Acikgoz M, Erdal D, Araci S: A new approach to q -Bernoulli numbers and q -Bernoulli polynomials related to q -Bernstein polynomials. Adv. Differ. Equ. 2010., 2010: Article ID 951764Google Scholar
- Araci S, Erdal D, Seo J: A study on the fermionic p -adic q -integral representation on associated with weighted q -Bernstein and q -Genocchi polynomials. Abstr. Appl. Anal. 2011., 2011: Article ID 649248Google Scholar
- Bayad A: Modular properties of elliptic Bernoulli and Euler functions. Adv. Stud. Contemp. Math. 2010, 20(3):389–401.MathSciNetMATHGoogle Scholar
- Bayad A, Kim T: Identities for the Bernoulli, the Euler and the Genocchi numbers and polynomials. Adv. Stud. Contemp. Math. 2010, 20(2):247–253.MathSciNetMATHGoogle 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
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