Sums of finite products of Legendre and Laguerre polynomials

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1 Kim et al. Advances in Difference Equations 28 28:277 R E S E A R C H Open Access Sums of finite products of Legendre and Laguerre polynomials Taeyun Kim,DaeSanKim 2,DmitryV.Dolgy 3 and Jin-Woo Par 4* * Correspondence: a47@nu.ac.r 4 Department of Mathematics Education, Daegu University, Gyeongsan-si, Republic of Korea Full list of author information is available at the end of the article Abstract In this paper, we study sums of finite products of Legendre and Laguerre polynomials and derive Fourier series epansions of functions associated with them. From these Fourier series epansions, we are going to epress those sums of finite products as linear combinations of Bernoulli polynomials. Further, by using a method other than Fourier series epansions, we will be able to epress those sums in terms of Euler polynomials. MSC: B68; 33C45; 42A6 Keywords: Fourier series; Finite products; Legendre polynomials; Laguerre polynomials; Bernoulli polynomials; Euler polynomials Introduction and preliminaries The Coulomb potential can be written as a series in Legendre polynomials P n n. They satisfy the orthogonality relation P n P m d 2 2n + δ n,m, and are solutions to the Legendre equation 2 P n 2P n +nn +P n. We let the reader refer to [] for further applications of Legendre polynomials. The Laguerre polynomials L n have important applications to the solution of Schrödinger s equation for the hydrogen atom. They are orthogonal over the interval [, with the weight function e,namely L n L m e d δ n,m, and are solution to the Laguerre equation L n + L n +nl n. The Authors 28. This article is distributed under the terms of the Creative Commons Attribution 4. International License which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original authors and the source, provide a lin to the Creative Commons license, and indicate if changes were made.

2 Kim etal. Advances in Difference Equations 28 28:277 Page 2 of 7 The Legendre polynomials P n and Laguerre polynomials L n are respectively defined by the recurrence relations as in the following see [2 4]: n +2P n+2 2n +3P n+ n +P n n, P, P, n +2L n+2 2n +3 L n+ n +L n n, L, L Both P n andl n are polynomials of degree n with rational coefficients. From. and.2, it can be easily seen that the generating functions for P n and L n aregivenbysee[2 4] Ft, 2t + t 2 2 P n t n,.3 n Gt, t ep t t L n t n..4 As is well nown, the Bernoulli polynomials B n aregivenby t e t et n For any real number,we let n B n tn n!..5 [] [,.6 denote the fractional part of,where[] is the greatest integer. We recall here that a for m 2, B m m! n n e m ;.7 b n n e B, for R Z,, for Z..8 For any integers m, r with m, r, we put α m,r P i P i2 P i2r+,.9 i +i 2 + +i 2r+ m where the sum runs over all nonnegative integers i, i 2,...,i 2r+ with i + i i 2r+ m.

3 Kim etal. Advances in Difference Equations 28 28:277 Page 3 of 7 Then we will consider the function α m,r and derive its Fourier series epansions. As an immediate corollary to these Fourier series epansions, we will be able to epress α m,r as a linear combination of Bernoulli polynomials B n.westatethishereastheorem A. Theorem A For any integers m, rwithm, r, we let m,r 2r!!2 m+r Then we have the identity i + +i 2r+ m [ m 2 ] P i P i2r+ m + r 2r 2m +2r 2 m + r m j m + r 2 r. 2r +2j 3!! m j+,r+j B j.. 2r 3!!j! Here r r +for r,,2n!! 2n 2n 3 for n, and!!. Also, for any integers m, rwithm, r, we put β m,r i + +i r+ m L i L i2 r + r + L ir+ r +,. where the sum is over all nonnegative integers i, i 2,...,i r+ with i + + i r+ m. Then we will study the function β m,r and obtain its Fourier series epansions. Again, as a corollary to these, we can epress β m,r in terms of Bernoulli polynomials. Here our result is as follows. Theorem B For any integers m, rwithm, r, we let m m,r m m! Then the following identity holds: i + +i r+ m L i L i2 r + m + r r +. L ir+ r + m j m j+,r+j B j..2 j! j Here we note that neither P n norl n is an Appell polynomial, whereas all our related results, ecept [5], have been onlyabout Appell polynomials see [6 9]. Assume that the polynomials p n, q n, and r n havedegreen. Thelinearization problem in general consists in determining the coefficients c nm intheepansionofthe product of two polynomials q n andr m in terms of an arbitrary polynomial sequence

4 Kim etal. Advances in Difference Equations 28 28:277 Page 4 of 7 {p } : n+m q n r m c nm p. Thus our results in Theorems A and B can be viewed as generalizations of the linearization problem. Our study of sums of finite products of special polynomials in this paper can be further justified by the following. Let us put m γ m m B B m m 2..3 Just as we see in.and.2, γ m can be epressed in terms of Bernoulli polynomials from the Fourier series epansions of γ m. From these epansions and after some simple modification, we can obtain the famous Faber Pandharipande Zagier identity see [] and a variant of Mii s identity see [ 4]. The papers [5, 6] are ecellent sources for operational techniques. Finally, for some of the recent results, we let the reader refer to the papers [5 9, 7 9]. 2 Fourier series epansions for functions associated with Legendre polynomials We start with the following lemma which will play an important role in this section. Lemma 2. Let n, r be integers with n, r. Then we have the identity i +i 2 + +i 2r+ n P i P i2 P i2r+ 2r!! Pr n+r, 2. where the sum is over all nonnegative integers i, i 2,...,i 2r+ with i + i i 2r+ n. Proof By differentiating.3 r times, we obtain r Ft, r r Ft, r 2r!!t r 2t + t 2 r 2, 2.2 nr P r n tn n P r n+r tn+r. 2.3 From 2.2and2.3, we have 2r!! 2t + t 2 r+ 2 n P r n+r tn. 2.4

5 Kim etal. Advances in Difference Equations 28 28:277 Page 5 of 7 On the other hand, from.3and2.2weobservethat n i +i 2 + +i 2r+ n 2r+ P n t n n 2t + t 2 r 2 2r!! n P i P i2 P i2r+ t n P r n+r tn. 2.5 Comparing both sides of 2.5, we get the desired result. It is nown that the Legendre polynomials P n aregivenbysee[2, 3] P n 2 F n,n+ [ n 2 ] n 2n 2 n 2, n n where 2 F a,b c z a n b n z n n c n is the Gauss hypergeometric function with n! n denoting the rising factorial polynomial defined by n + + n n,. The rth derivative of 2.6isgivenby P r n 2 n n 2n 2 n [ n r 2 ] Combining 2.and2.7, we get the followinglemma. n 2 r n 2 r r n. 2.7 Lemma 2.2 For integers n, r with n, r, we have the following identity: i +i 2 + +i 2r+ n 2r!!2 n+r P i P i2 P i2r+ [ n 2 ] n + r 2n +2r 2 n + r For integers m, r with m, r asin.9, we let α m,r i +i 2 + +i 2r+ m Then we will consider the function α m,r i +i 2 + +i 2r+ m P i P i2 P i2r+. n + r 2 r n P i Pi2 Pi2r+, 2.9

6 Kim etal. Advances in Difference Equations 28 28:277 Page 6 of 7 defined on R, which is periodic with period. The Fourier series of α m,r is where n A m,r n A m,r n e, 2. α m,r e d For integers m, r, we put α m,r e d. 2. m,r α m,r α m,r Pi P i2 P i2r+ P i P i2 P i2r i + +i 2r+ m Now, from 2.8, we see that m,r 2r!!2 m+r [ m 2 ] m + r 2m +2r 2 m + r m + r 2 r. 2.3 Here we note that m 2 m+r m+2r 2r!!2 α m,r m+r m 2 m+r r!, if m even,, if m odd. 2.4 From 2., we observe that d d α m,r d d 2r!! Pr m+r 2r!! Pr+ m+r 2r +α m,r+. Thus we have shown that d d α m,r2r +α m,r Replacing m by m +,r by r,from2.5weget d d 2r α m+,r α m,r, 2.6 α m,r d 2r m+,r, 2.7

7 Kim etal. Advances in Difference Equations 28 28:277 Page 7 of 7 α m,r α m,r m,r. 2.8 We are now ready to determine the Fourier coefficients A m,r n. Case : n. A m,r n α m,r e d [ αm,r e ] + αm,r α m,r 2r + + 2r + Am,r+ 2r + d d α m,r e d n m,r 2r +3 Am 2,r+2 n m,r+ 2r + 3!! 2 2r!! Am 2,r+2 n Case 2: n. A m,r 2r +2m!! m 2r!! A,r+m n 2r m j α m,r d 2 j m j α m,r+ e d m,r 2r +2j 3!! j 2r!! m j+,r+j 2r +2j 3!! j 2r!! m j+,r+j 2r +2j 3!! j 2r 3!! m j+,r+j r m+,r. 2.2 Now, from.7,.8, 2., 2., 2.9, and 2.2, we have the following Fourier series epansion of α m,r : 2r m+,r n n 2r 2r m+,r + 2r m j m j 2r +2j 3!! j 2r 3!! m j+,r+j e 2r +2j 3!! 2r 3!!j! 2r m+,r + 2r m j2 B, for / Z, + m,r, for Z m j+,r+j j! n n e j 2r +2j 3!! m j+,r+j B j 2r 3!!j!

8 Kim etal. Advances in Difference Equations 28 28:277 Page 8 of 7 2r m j j 2r +2j 3!! m j+,r+j B j 2r 3!!j! B, for / Z, + m,r, for Z. 2.2 α m,r m, r is piecewise C.Moreover,α m,r is continuous for those positive integers m, r with m,r, and discontinuous with jump discontinuities at integers for the positive integers m, r with m,r.thus,for m,r, the Fourier series of α m,r converges uniformly to α m,r, whereas, for m,r, the Fourier series of α m,r converges pointwise to α m,r for / Z and converges to αm,r + α m,r α m,r m,r m 2 m+r m+2r 2r!!2 m+r m 2 m+r r!+ 2 m,r, ifm even, 2 m,r, ifm odd, 2.22 for Z see 2.4. From these observations together with 2.2 and2.22, we obtain the net two theorems. Theorem 2.3 For any integers m, rwithm, r, we let m,r 2r!!2 m+r [ m 2 ] m + r 2m +2r 2 m + r m + r 2 r. Assume that m,r for some positive integers m, r. Then we have the following: a P i Pi2 Pi2r+ i +i 2 + +i 2r+ m has the Fourier series epansion P i Pi2 Pi2r+ i +i 2 + +i 2r+ m 2r m+,r 2r m n n j 2r +2j 3!! j 2r 3!! m j+,r+j e b for all R, where the convergence is uniform. i +i 2 + +i 2r+ m 2r P i Pi2 Pi2r+ m j j 2r +2j 3!! m j+,r+j B j 2r 3!!j! for all R.

9 Kim etal. Advances in Difference Equations 28 28:277 Page 9 of 7 Theorem 2.4 For any integers m, rwithm, r, we let m,r 2r!!2 m+r [ m 2 ] m + r 2m +2r 2 m + r m + r 2 r. Assume that m,r for some positive integers m, r. Then we have the following: a b 2r m+,r 2r m n n j 2r +2j 3!! j 2r 3!! m j+,r+j e i + +i 2r+ m P i P i2 P i2r+, for / Z, m 2 m+r m+2r 2r!!2 m+r m 2 m+r r!+ 2 m,r, for Z and m even, 2 m,r, for Z and m odd. 2r 2r for Z. m j 2r +2j 3!! m j+,r+j B j 2r 3!!j! i +i 2 + +i 2r+ m m j j P i Pi2 Pi2r+, for / Z; 2r +2j 3!! m j+,r+j B j 2r 3!!j! m 2 m+r m+2r 2r!!2 m+r m 2 m+r r!+ 2 m,r, if m even, 2 m,r, if m odd, Finally, we observe that the statement in Theorem A follows immediately from Theorems 2.3 and Fourier series epansions for functions associated with Laguerre polynomials The following lemma is needed for our discussion in this section. Lemma 3. Let n, r be integers with n, r. Then we have the identity i +i 2 + +i r+ n L i L i2 r + r + L ir+ r + r L r n+r, 3. where the sum runs over all nonnegative integers i, i 2,...,i r+ with i + i i r+ n. Proof Differentiating.4 r times, we get r Gt, r r t r t r ep t, 3.2 t

10 Kim etal.advances in Difference Equations 28 28:277 Page of 7 r Gt, r nr L r n tn Equating 3.2and3.3gives us n r t r ep t t L r n+r tn+r. 3.3 n On the other hand, from.4and3.4, we note that n i +i 2 + +i r+ n L i L i2 r + r+ L n t n r + n t t ep r + t r n L r n+r tn. 3.4 r + r+ L ir+ r + t n L r n+r tn. 3.5 Comparing both sides of 3.5 yields the desired result. It is nown that the Laguerre polynomials L n aregivenbysee[2, 3] L n F n n n!, 3.6 where F a b z a n z n n b n is the hypergeometric function. n! The rth derivative of 3.6isgivenby n n n r r! L r r n r n r r! n r +r n r n. 3.7! + r From 3.and3.7, we obtain the following result. Lemma 3.2 For integers n, r with n, r, we have the following identity: i +i 2 + +i r+ n L i L i2 r + r + L ir+ r + n n + r. 3.8! + r

11 Kim etal.advances in Difference Equations 28 28:277 Page of 7 For integers m, r with m, r asin., we put β m,r i +i 2 + +i r+ m L i L i2 r + Then we will consider the function β m,r i +i 2 + +i r+ m r + defined on R, which is periodic with period. The Fourier series of β m,r is where n B m,r n L ir+ r + L i L i2 L ir+, 3.9 r + r + r + B m,r n e, 3. β m,r e d β m,r e d. 3. For integers m, r with m, r, we set. m,r β m,r β m,r L i L i2 L ir+ r + r + r + i +i 2 + +i r+ m L i L i2 L ir Now, from 3.8, we see that m m,r m m! From 3., we note that m + r d d β m,r d r L r m+r d r L r+ m+r β m,r Thus we have shown that d d β m,r β m,r+. 3.4

12 Kim etal.advances in Difference Equations 28 28:277 Page 2 of 7 Replacing m by m +,r by r,from3.4, we get d βm+,r β m,r, 3.5 d β m,r d m+,r, 3.6 β m,r β m,r m,r. 3.7 We are now going to determine the Fourier coefficients. Case : n. B m,r n β m,r e d [ βm,r e ] + βm,r β m,r d d β m,r e d β m,r+ e d Bm,r+ n m,r Bm 2,r+2 n m,r+ m,r 2 2 B m 2,r+2 n + j m j+,r+j m m B,r+m n + j m j+,r+j j j m j m j+,r+j. 3.8 j Case 2: n. B m,r β m,r d m+,r. 3.9 Now, from.7,.8, 3., 3., 3.8, and 3.9, the Fourier series epansion of β m,r isgivenby m+,r + m n n m+,r m+,r j j m j+,r+j e m j m j+,r+j j! j! j n n m j m j+,r+j B j j! j2 e j

13 Kim etal.advances in Difference Equations 28 28:277 Page 3 of 7 B, if / Z, + m,r, if Z m j B, if / Z, m j+,r+j B j + m,r j!, if Z. j j 3.2 β m,r m, r is piecewise C. In addition, β m,r is continuous for those integers m, r m, r with m,r, and discontinuous with jump discontinuities at integers for those integers m, r m, r with m,r.hence,for m,r,the Fourier series of β m,r converges uniformly to β m,r. Whereas, for m,r,the Fourier series of β m,r convergespointwisetoβ m,r for / Z and converges to βm,r + β m,r β m,r + m + r 2 2 m,r + r 2 m,r 3.2 for Z see 3.8. Fromthese observations together with 3.2and3.2, we have the net two theorems. Theorem 3.3 For any integers m, rwithm, r, we let m m,r m m! m + r. Assume that m,r for some integers m, rwithm, r. Then we have the following: a i +i 2 + +i r+ m has the Fourier series epansion i +i 2 + +i r+ m m+,r + L i L i2 L ir+ r + r + r + L i L i2 L ir+ r + r + r + m n n j j m j+,r+j e b for all R, where the convergence is uniform. i +i 2 + +i r+ m L i L i2 L ir+ r + r + r + m j m j+,r+j B j j! j j for all R.

14 Kim etal.advances in Difference Equations 28 28:277 Page 4 of 7 Theorem 3.4 For any integers m, rwithm, r, we let m m,r m m! m + r. Assume that m,r for some integers m, rwithm, r. Then we have the following: a m+,r + m n n j j m j+,r+j e i +i 2 + +i r+ m L i r+ L i 2 r+ L i r+, for / Z, r+ m+r r + 2 m,r, for Z. b m j m j+,r+j B j j! j m j j i +i 2 + +i r+ m L i r + m + r r L i L i2 L ir+, for / Z; r + r + r + L i2 r m,r, for Z. L ir+ r + Finally, we see that the statement in Theorem B follows from Theorems 3.3 and Epressions in terms of Euler polynomials For any polynomial p C[] with degree m, we now that m p b E, 4. where E are the Euler polynomials given by 2 e t + et E t!,and b p + p,,,...,m ! Applying 4.and4.2topα m,r andfrom2.5, we see that α m,r 2r +2!! α m,r r!!

15 Kim etal.advances in Difference Equations 28 28:277 Page 5 of 7 Hence, from 4.3, we have b α m,r + α m,r 2! 2r +2!! 2!2r!! 2r +2!!!2r!! αm,r+ + α m,r+ α m,r+ + 2 m,r Now, from 2.4, 4., and 4.4, we get the following theorem. Theorem 4. For any integers m, rwithm, r, we have the following: m P i P i2 P i2r+ b E, where i +i 2 + +i 2r+ m b m 2 r+! m+r!2r!!2 m+r m 2 2r+2!! 2!2r!! m,r+, m+2r+ m+r + 2r+2!! 2!2r!! m,r+, for m even, for m odd, and m,r is given by 2.3. Proceeding analogously to the above discussion, we get the net result, the details of which are left to the reader. Theorem 4.2 For any integers m, r with m, r,we have the following identity: i +i 2 + +i r+ m L i L i2 r + r + L ir+ m m + r +! r + 2 m,r+ E, where m,r is given by 3.3. r + 5 Results and discussion In this paper, we investigated sums of finite products of Legendre and Laguerre polynomials and derived Fourier series epansions of functions associated with those polynomials. FromtheseFourierseriesepansions,wewereabletoepressthosesumsoffiniteproducts as linear combinations of Bernoulli polynomials. Further, by using a different method, we epressed those sums of finite products as linear combinations of Euler polynomials as well. It is epected that we will be able to epress those sums of finite products as linear combinations of some classical orthogonal polynomials. Here we note that the Bernoulli and Euler polynomials are not orthogonal polynomials but Appell polynomials.

16 Kim etal.advances in Difference Equations 28 28:277 Page 6 of 7 6 Conclusion In this paper, we considered the Fourier series epansions for functions associated with Legendre and Laguerre polynomials. In addition, by using a method other than Fourier series epansions, we were able to epress those sums in terms of Euler polynomials. It is noteworthy that all the other previous related papers, ecept [5, 2 22], were associated with Appell polynomials, while the present one is about classical orthogonal polynomials. Funding This research was supported by the Basic Science Research Program through the National Research Foundation of Korea NRF funded by the Ministry of Education No. NRF-27RDAB Competing interests The authors declare that they have no competing interests. Authors contributions All authors contributed equally to the manuscript and typed, read, and approved the final manuscript. Author details Department of Mathematics, Kwangwoon University, Seoul, Republic of Korea. 2 Department of Mathematics, Sogang University, Seoul, Republic of Korea. 3 Hanrimwon, Kwangwoon University, Seoul, Republic of Korea. 4 Department of Mathematics Education, Daegu University, Gyeongsan-si, Republic of Korea. Publisher s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Received: 4 June 28 Accepted: 3 July 28 References. Cesarano, C., Ricci, P.E.: The Legendre polynomials as a basis for Bessel functions. Int. J. Pure Appl. Math., Andrews, G.E., Asey, R., Roy, R.: Special Functions. Encyclopedia of Mathematics and Its Applications, vol. 7. Cambridge University Press, Cambridge Beals, R., Wang, R.: Special Functions: A Graduate Tet. Cambridge Studies in Advanced Mathematics, vol. 26. Cambridge University Press, Cambridge 2 4. Dattoli, G., Srivastava, H.M., Cesarano, C.: The Laguerre and Legendre polynomials from an operational point of view. Appl. Math. Comput. 24, Kim, T., Kim, D.S., Dolgy, D.V., Par, J.W.: Sums of finite products of Chebyshev polynomials of the second ind and of Fibonacci polynomials. J. Inequal. Appl. 28, Agarwal, R.P., Kim, D.S., Kim, T., Kwon, J.: Sums of finite products of Bernoulli functions. Adv. Differ. Equ. 27, Kim, T., Kim, D.S., Jang, G.-W., Kwon, J.: Sums of finite products of Euler functions. In: Advances in Real and Comple Analysis with Applications. Trends in Math., pp Springer, Berlin Kim, T., Kim, D.S., Jang, G.-W., Kwon, J.: Fourier series of finite products of Bernoulli and Genocchi functions. J. Inequal. Appl. 27, Kim, T., Kim, D.S., Jang, L.C., Jang, G.-W.: Sums of finite products of Genocchi functions. Adv. Differ. Equ. 27, Faber, C., Pandharipande, R.: Hodge integrals and Gromov Witten theory. Invent. Math. 39, Dunne, G.V., Schubert, C.: Bernoulli number identities from quantum field theory and topological string theory. Commun. Number Theory Phys. 7, Gessel, I.M.: On Mii s identity for Bernoulli numbers. J. Number Theory, Mii, H.: A relation between Bernoulli numbers. J. Number Theory, Shiratani, K., Yooyama, S.: An application of p-adic convolutions. Mem. Fac. Sci., Kyushu Univ., Ser. A, Math. 36, Dattoli, G., Lorenzutta, S., Cesarano, C.: Bernstein polynomials and operational methods. J. Comput. Anal. Appl. 8, Dattoli, G., Lorenzutta, S., Ricci, P.E., Cesarano, C.: On a family of hybrid polynomials. Integral Transforms Spec. Funct. 5, Bayad, A., Navas, L.: Algebraic properties and Fourier epansions of two-dimensional Apostol Bernoulli and Apostol Euler polynomials. Appl. Math. Comput. 265, Simse, Y.: Combinatorial identities associated with Bernstein type basis functions. Filomat 37, Simse, Y.: Formulas for Poisson Charlier, Hermite, Milne Thomson and other type polynomials by their generating functions and p-adic integral approach. Rev. R. Acad. Cienc. Eactas Fís. Nat., Ser. A Mat., 8 to appear Kim, T., Kim, D.S., Dolgy, D.V., Ryoo, C.S.: Representing sums of finite products of Chebyshev polynomials of third and fourth inds by Chebyshev polynomials. Symmetry 7,258 28

17 Kim etal.advances in Difference Equations 28 28:277 Page 7 of 7 2. Kim, D.S., Kim, T., Kwon, H.-I., Kwon, J.: Fourier series of sums of products of higher-order Genocchi functions. Adv. Stud. Contemp. Math. Kyungshang 282, Kim, D.S., Kim, T., Kwon, H.-I.: Fourier series of r-derangement and higher-order derangement functions. Adv. Stud. Contemp. Math. Kyungshang 28, 28