Sums of finite products of Chebyshev polynomials of the third and fourth kinds

Transcription

1 Kim et al. Advances in Difference Equations 28 28:283 R E S E A R C H Open Access Sums of finite products of Chebyshev polynomials of the third and fourth kinds Taekyun Kim,2,DaeSanKim 3,DmitryV.Dolgy 4 and Jongkyum Kwon 5* * Correspondence: mathkjk26@gnu.ac.kr 5 Department of Mathematics Education and ERI, Gyeongsang National University, Jinju, 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 Chebyshev polynomials of the third and fourth kinds and obtain Fourier series expansions of functions associated with them. Then from these Fourier series expansions we will be able to express those sums of finite products as linear combinations of Bernoulli polynomials and to have some identities from those expressions. MSC: B83; 42A6; 42C Keywords: Fourier series; Chebyshev polynomials; Sums of finite products of Chebyshev polynomials of the third and fourth kinds Introduction and preliminaries The Chebyshev polynomials T n x, U n x, V n x, and W n x of the first, second, third, and fourth kinds are respectively defined by the recurrence relations as follows see [2, 6, 7, 2, 4]: T n+2 x2xt n+ x T n x n, T x, T xx,. U n+2 x2xu n+ x U n x n, U x, U x2x,.2 V n+2 x2xv n+ x V n x n, V x, V x2x,.3 W n+2 x2xw n+ x W n x n, W x, W x2x +..4 It can be easily seen from.,.2,.3, and.4 that the generating functions for T n x, U n x, V n x, and W n x are respectively given by see [6, 7, 2] xt 2xt + t T 2 n xt n,.5 n 2xt + t U 2 xt n,.6 n t Ft, x 2xt + t V 2 n xt n,.7 n +t Gt, x 2xt + t W 2 n xt n..8 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 link to the Creative Commons license, and indicate if changes were made.

2 Kim etal. Advances in Difference Equations 28 28:283 Page 2 of 7 The Bernoulli polynomials B m x are given by the generating function t e t ext B m x tm m!..9 m For any real number x,we let x [x] [,. denote the fractional part of x,where[x] indicates the greatest integer x. We also recall here that a for m 2, B m m! n,n e 2πinx 2πin m ;. b for m, n,n e 2πinx 2πin B, for x R Z,, for x Z..2 For any integers m, r with m, r, we set α m,r x +m l V c x V cr+ x,.3 where the inner sum runs over all nonnegative integers c, c 2,...,c r+ with c + c c r+ l. Then we will consider the function α m,r and derive its Fourier series expansions. As an immediate corollary to these Fourier series expansions, we will be able to express α m,r x as a linear combination of Bernoulli polynomials B m x. We state our result here as Theorem.. Theorem. For any integers m, rwithm, r, we let m,r r! k Then we have the identity j k+ m + + k +2k +m l k + r r 2 k. V c x V cr+ x r + j 2 j m j+,r+j B j x..4 Here x r xx x r +,for r, and x.

3 Kim etal. Advances in Difference Equations 28 28:283 Page 3 of 7 Also, for any integers m, r with m, r, we put β m,r x +m l m l W c x W cr+ x,.5 where the inner sum is over all nonnegative integers c, c 2,...,c r+ with c +c 2 + +c r+ l. Then we will consider the function β m,r and derive its Fourier series expansions. Again, as a corollary to these, we can express β m,r x in terms of Bernoulli polynomials. Indeed, our result here is as follows. Theorem.2 For any integers m, rwithm, r, we let m,r 2m + + r! Then we have the identity j k k+ 2 k 2k + + m + + k 2k + +m l m l W c x W cr+ x k + r r. r + j 2 j m j+,r+j B j x..6 Here we cannot go without saying that neither V n xnorw n x is an Appell polynomial, whereas all our related results have been only about Appell polynomials see [, 8 ]. We mentioned in the above that, using the Fourier series expansions of α m,r and β m,r, we can express α m,r x andβ m,r x as linear combinations of Bernoulli polynomialsasstatedintheorem. and Theorem.2. In addition, we will express α m,r x andβ m,r x as linear combinations of Euler polynomials by using a simple formula see 4., 4.3. Finally, as an application of our results, we will derive some interesting identities from Theorem. and Theorem.2 together with some well-known identities connecting the four kinds of Chebyshev polynomials see It was mentioned in [] that studying these kinds of sums of finite products of special polynomials can be well justified by the following example. Let us put m γ m x km k B kxb m k x m 2..7 k Then just as in [4] and[6] wecanexpressγ m x in terms of Bernoulli polynomials from the Fourier series expansions of γ m. Further, some simple modification of this gives the famous Faber Pandharipande Zagier identity see [4] and a slightly different variant of Miki s identity see [3, 5, 3, 5]. Our methods here are simple, whereas others are quite involved. Indeed, Dunne and Schubert in [3] use the asymptotic expansions of some special polynomials coming from quantum field theory computations, the work of Gessel in [5] is based on two different

4 Kim etal. Advances in Difference Equations 28 28:283 Page 4 of 7 expressions for the Stirling numbers of the second kind, Miki in [3]utilizesaformulafor the Fermat quotient ap a modulo p 2, and Shiratani and Yokoyama in [5] employp-adic p analysis. The reader may refer to the papers [, 8 ]forsomerelatedrecentresults. 2 Fourier series expansions for functions associated with the Chebyshev polynomials of the third kind The following lemma, which expresses the sums of products in.3 neatly,willplaya crucial role in this section. The corresponding ones for Chebyshev polynomials of the second kind and of Fibonacci polynomials are respectively stated in [6]and[7]. Lemma 2. Let n, r be integers with n, r. Then we have the identity n +n l V c x V cr+ x 2 r r! V r n+r x, where the inner sum runs over all nonnegative integers c, c 2,...,c r+ with c + c c r+ l. Proof By differentiating.7 r times, we have r Ft, x 2tr r! t, x r 2xt + t 2 r+ 2. r Ft, x V r x r n xtn V n+r r. 2.2 nr n Equating 2.and2.2gives 2 r r! t 2xt + t 2 V r r+ n+r xtn. 2.3 n On the other hand, using.7and 2.3wenotethat n c +c 2 + +c r+ n r+ V n xt n n t r+ 2xt + t 2 r+ 2 r tr r! V c x V cr+ x t n n V r n+r xtn. 2.4

5 Kim etal. Advances in Difference Equations 28 28:283 Page 5 of 7 From 2.4, we obtain 2 r r! n t r V r n+r xtn m c +c 2 + +c r+ m r + m t m m n n l V c x V cr+ x t m c +c 2 + +c r+ l r + n l V c x V cr+ x t l V c x V cr+ x t n, 2.5 from which our result follows. It is well known that the Chebyshev polynomials of the third kind V n x aregivenby see [6] V n x 2 F n, n +; 2 ; x 2 n n + k 2 k x k, 2.6 2k k where 2 F a, b; c; z is the hypergeometric function. The rth derivative of 2.6isgivenby n V n r x kr n + k 2k 2 k k r x k r r n. 2.7 Combining 2.and2.7, we obtain thefollowinglemma. Lemma 2.2 For integers n, r, with n, r, we have the following identity: n r! k +n l V c x V cr+ x n n + + k 2 k k + r r x k k For integers m, r with m, r, as in.3, we let α m,r x +m l Then we will consider the function m +m l α m,r defined on R, which is periodic with period. V c x V cr+ x. 2.9 V c Vcr+, 2.

6 Kim etal. Advances in Difference Equations 28 28:283 Page 6 of 7 The Fourier series of α m,r is where n A m,r n A m,r n e 2πinx, 2. For m, r, we let α m,r e 2πinx dx α m,r xe 2πinx dx. 2.2 m,r α m,r α m,r +m l V c V cr+ V c V cr Then, from 2.8and2.3,we see that m,r r! k where we observe that k+ m + + k +2k 2 k k + r r, 2.4 m + α m,r. 2.5 Now, from 2., we note the following: d dx α m,rx d dx 2 r r! V m+r r x 2 r r! V m+r r+ x +α m,r+ x. 2.6 Thus we have shown that d dx α m,rx +α m,r+ x. 2.7 Replacing m by m +,r by,from2.7weobtain d αm+,r x α m,r x, 2.8 dx α m,r x dx m+,r, 2.9 α m,r α m,r m,r. 2.2

7 Kim etal. Advances in Difference Equations 28 28:283 Page 7 of 7 We are now going to determine the Fourier coefficients A m,r n. Case : n. A m,r n α m,r xe 2πinx dx [ αm,r xe 2πinx] 2πin + 2πin αm,r α m,r + + 2πin 2πin + 2πin Am,r+ + 2πin d dx α m,rx e 2πinx dx n 2πin m,r +2 2πin Am 2,r+2 n 2πin m,r+ 22 r πin 2 Am 2,r+2 n 2m r + m m 2πin m A,r+m n j Case 2: n. A m,r 2 j j 2 j r + j j 2πin j m j+,r+j j α m,r+ xe 2πinx dx 2πin m,r 2 j r + j j 2πin j m j+,r+j 2 j r + j j 2πin j m j+,r+j 2 j r + j j 2πin j m j+,r+j. 2.2 α m,r x dx m+,r From.,.2, 2.2, and 2.22, we get the Fourier series of α m,r as follows: m+,r n,n m+,r + j! n,n m+,r + j j e 2πinx 2πin j j2 2 j r + j j 2πin j m j+,r+j 2 j r + j B, for x R Z, + m,r, for x Z m j+,r+j r + j 2 j m j+,r+j B j e 2πinx

8 Kim etal. Advances in Difference Equations 28 28:283 Page 8 of 7 j,j r + j 2 j m j+,r+j B j B, for x R Z, + m,r, for x Z 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 those 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 x R Z,andconvergesto αm,r + α m,r α m,r 2 2 m,r m + 2 m,r, forx Z see From these observations, together with 2.23 and2.24, we obtain the next two theorems. Theorem 2.3 For any integers m, rwithm, r, we let m,r r! k k+ m + + k +2k 2 k k + r r. Assume that m,r for some positive integers m, r. Then we have the following. a m l Vc V cr+ has the Fourier series expansion c +c 2 + +c r+ l m+,r r +m l r +m l n,n V c Vcr+ j 2 j r + j j 2πin j m j+,r+j e 2πinx, 2.25 b for all x R, where the convergence is uniform. for all x in R. j,j +m l 2 j r + j V c Vcr+ m j+,r+j B j 2.26 Theorem 2.4 For any integers m, rwithm, r, we let m,r r! k k+ m + + k +2k 2 k k + r r.

9 Kim etal. Advances in Difference Equations 28 28:283 Page 9 of 7 Assume that m,r for some positive integers m, r. Then we have the following. a m+,r n,n j 2 j r + j j 2πin j m j+,r+j e 2πinx m r +m l l c +c 2 + +c r+ l Vc V r cr+, for x / Z, m+ 2 m,r, for x Z b j r + j 2 j m j+,r+j B j +m l for x R Z; r + j 2 j m j+,r+j B j j,j m + 2 m,r, for x Z. V c Vcr+, 2.28 Finally,weobservethat,fromTheorems2.3 and 2.4, we immediately obtain the result in Theorem. expressing α m,r x in terms of Bernoulli polynomials. 3 Fourier series expansions for functions associated with the Chebyshev polynomials of the fourth kind Here we will omit the details for the results in this section, as all of them can be obtained analogously to the previous section. We start with the following important lemma. Lemma 3. Let n, r be integers with n, r. Then we have the following identity: n +n l n l W c x W cr+ x 2 r r! W r n+r x, where the inner sum runs over all nonnegative integers c, c 2,...,c r+ with c + c c r+ l. As is well known, the Chebyshev polynomials of the fourth kind W n x are explicitly given by see [6] W n x2n + 2 F n, n +; 3 2 ; x 2 2n + n k 2 k 2k + n + k 2k where 2 F a, b; c; z is the hypergeometric function. x k, 3.

10 Kim etal.advances in Difference Equations 28 28:283 Page of 7 The rth derivative of 3.isgivenby n W n r x2n + 2 k 2k + kr n + k 2k Combining 3.and3.2 yields thefollowinglemma. k r x k r r n. 3.2 Lemma 3.2 For integers n, r with n, r, we have the following identity: n r! 2n + + n +n l n l W c x W cr+ x k As in.5, for m, r, we let 2 k 2k + + n + + k 2k + k + r r x k. 3.3 β m,r x +m l m l W c x W cr+ x. 3.4 Then the Fourier series of β m,r is where n B m,r n B m,r n e 2πinx, 3.5 For m, r, we put β m,r e 2πinx dx β m,r xe 2πinx dx. 3.6 m,r β m,r β m,r +m l m l W c W cr+ W c W cr Then, from 3.3and3.7, we have m,r 2m + + r! wherewenotethat β m,r 2m k k+ 2 k 2k + + m + m + + k 2k + k + r r,

11 Kim etal.advances in Difference Equations 28 28:283 Page of 7 From 3., we can deduce the following: d dx β m,rx +β m,r+ x, 3. d βm+,r x β m,r x, 3. dx β m,r x dx m+,r, 3.2 β m,r β m,r m,r. 3.3 The Fourier series expansion of β m,r is as follows: m+,r j,j n,n j 2 j r + j j 2πin j m j+,r+j r + j 2 j m j+,r+j B j B, for x R Z, + m,r, for x Z. e 2πinx 3.4 We also observe that βm,r + β m,r β m,r 2 2 m,r 2m m + Now, the next two theorems follow from 3.4and m,r. 3.5 Theorem 3.3 For any integers m, rwithm, r, we let m,r 2m + + r! k k+ 2 k 2k + + m + + k 2k + k + r r. Assume that m,r for some positive integers m, r. Then we have the following. a m l Wc W cr+ has the Fourier series expansion c +c 2 + +c r+ l m l r +m l r m+,r +m l m l W c Wcr+ n,n for all x R, where the convergence is uniform. j 2 j r + j j 2πin j m j+,r+j e 2πinx, 3.6

12 Kim etal.advances in Difference Equations 28 28:283 Page 2 of 7 b j,j +m l m l W c Wcr+ 2 j r + j m j+,r+j B j 3.7 for all x in R. Theorem 3.4 For any integers m, rwithm, r, we let m,r 2m + + r! k k+ 2 k 2k + + m + + k 2k + k + r r. Assume that m,r for some positive integers m, r. Then we have the following: a m+,r n,n j 2 j r + j j 2πin j m j+,r+j e 2πinx m l c +c 2 + +c r+ l m l r +m l Wc W r cr+, for x R Z, 2m m,r, for x Z. m+ 3.8 b j r + j 2 j m j+,r+j B j +m l m l W c Wcr+, for x R Z; 3.9 r + j 2 j m j+,r+j B j j,j 2m m + 2 m,r, for x Z. Now, as an immediate corollary to Theorems 3.3 and 3.4, wegetthestatedresultin Theorem.2 expressing β m,r x in terms of Bernoulli polynomials. 4 Expressions in terms of Euler polynomials Let px C[x] be a polynomial of degree m.thenitisknownthat px b k E k x, 4. k

13 Kim etal.advances in Difference Equations 28 28:283 Page 3 of 7 where E k x are the Euler polynomials given by and 2 e t + ext k E k x tk k!, 4.2 b k p k + p k, k,,...,m k! Applying 4.and4.3topxα m,r x, from 2.7wehave α k m,r x2k r + k k α m k,r+k x. 4.4 Hence, from 4.4 we see that b k α k m,r + αk m,r 2k! r + k 2 αm k,r+k k + α m k,r+k r 2 k r + k r Now, we note from 2.5that 2αm k,r+k m k,r+k. 4.5 m + + k α m k,r+k k Combining 4.5and4.6, we finallyobtain r + k b k 2 k 2 r m + + k +2k Thus we have the following theorem from 4.and4.7. m k,r+k. 4.7 Theorem 4. For any integers m, r with m, r, we have the following: k 2 k r + k r +m l where m,r is given by 2.4. m + + k 2 +2k V c x V cr+ x m k,r+k E k x, 4.8 The next theorem follows analogously to the previous discussion and hence the details will be left to the reader.

14 Kim etal.advances in Difference Equations 28 28:283 Page 4 of 7 Theorem 4.2 For any integers m, r with m, r, we have the following: k 2 k r + k r where m,r is given by m l m l W c x W cr+ x 22m + + m + + k +2k + +2k m k,r+k E k x, Applications Let T n x, U n xn be the Chebyshev polynomials of the first kind and of the second kind respectively given by.or.5, and.2or.6. The statement in the following lemma is from equations.2and.3of[2]. Lemma 5. Let u [ 2 + x] 2. Then we have the following identities: W n x n V n x, 5. V n xu T 2n+ u, 5.2 W n xu 2n u. 5.3 From.4and5.2andwithu as in Lemma 5.,wehave ur+ m j +m l T 2c +u T 2cr+ +u r + j 2 j m j+,r+j B j x. 5.4 In particular, for x,5.4yields 2 r+3 2 r j +m l T 2c + 2 T 2cr+ + 2 r + j 2 j m j+,r+j B j. 5.5 On the other hand, from.6and5.3andwithu as in Lemma 5.,weget j +m l m U 2c u U 2cr+ u r + j 2 j m j+,r+j B j x. 5.6

15 Kim etal.advances in Difference Equations 28 28:283 Page 5 of 7 Now, letting x in5.6gives j +m l 2 2 U m 2c U 2cr+ r + j 2 j m j+,r+j B j. 5.7 Finally, from.4,.6, and 5., we obtain j m m +m l l W c x W cr+ x r + j 2 j m j+,r+j B j x j In particular, when x,5.8 gives j r + j 2 j m j+,r+j B j m m j r + j 2 j m j+,r+j B j x. 5.8 r + j 2 j m j+,r+j B j. 5.9 In addition, from 5.8 and using the well-known fact that B j x j B j x+jx j, we have r + j 2 j j m j+,r+j x j j m r + j m 2 m j+,r+j j + m j+ m j+,r+j Bj x, 5. j where we observed that,r+m,r+m Results and discussion Let α m,r x, β m,r x denote the following sums of finite products given by α m,r x β m,r x +m l V c x V cr+ x, +m l m l W c x W cr+ x, 6.

16 Kim etal.advances in Difference Equations 28 28:283 Page 6 of 7 where V n x, W n x n are respectively the Chebyshev polynomials of the third kind and of the fourth kind. Then we considered the functions α m,r, β m,r whichare obtained by extending by periodicity from α m,r x, β m,r x on [,. We obtained the Fourier series expansions of α m,r, β m,r, and expressed α m,r x, β m,r x aslinear combinations of the usual Bernoulli polynomials. Moreover, we expressed α m,r x, β m,r x in terms of the usual Euler polynomials and derived several identities involving the four kinds of Chebyshev polynomials. We expect that we can get similar results for some other orthogonal polynomials. 7 Conclusion In this paper, we studied sums of finite products of Chebyshev polynomials of the third kind and of the fourth kind. In recent years, we have obtained similar results for many other special polynomials. However, all of them have been the Appell polynomials, whereas Chebyshev polynomials are the classical orthogonal polynomials. Studying these kinds of sums of finite products of special polynomials can be well justified by the following example. Let us put m γ m x km k B kxb m k x m k Then we can express γ m x in terms of Bernoulli polynomials from the Fourier series expansions of γ m, just as we did these for α m,r x, β m,r x. Further, some simple modification of this gives us the famous Faber Pandharipande Zagier identity and a slightly different variant of Miki s identity. We note here that all the other known derivations of Faber Pandharipande Zagier identity and Miki s identity are quite involved. But our approach to 7. via Fourier series is elementary but powerful enough to get many results. We expect that we can get similar results for some other orthogonal polynomials. Acknowledgements The authors would like to express their sincere gratitude to the editor and referees, who gave us valuable comments to improve this paper. The first author s work in this paper was conducted during the sabbatical year of Kwangwoon University in 28. Funding This work was supported by the National Research Foundation of Korea NRF grant funded by the Korea government MEST No. 27REAA 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, College of Science, Tianjin Polytechnic University, Tianjin, China. 2 Department of Mathematics, Kwangwoon University, Seoul, Republic of Korea. 3 Department of Mathematics, Sogang University, Seoul, Republic of Korea. 4 Hanrimwon, Kwangwoon University, Seoul, Republic of Korea. 5 Department of Mathematics Education and ERI, Gyeongsang National University, Jinju, Republic of Korea. Publisher s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Received: 8 May 28 Accepted: 3 July 28

17 Kim etal.advances in Difference Equations 28 28:283 Page 7 of 7 References. Agarwal, R.P., Kim, D.S., Kim, T., Kwon, J.: Sums of finite products of Bernoulli functions. Adv. Differ. Equ. 27, Doha, E.H., Abd-Elhameed, W.M., Alsuyuti, M.M.: On using third and fourth kinds Chebyshev polynomials for solving the integrated forms of high odd-order linear boundary value problems. J. Egypt. Math. Soc. 23, Dunne, G.V., Schubert, C.: Bernoulli number identities from quantum field theory and topological string theory. Commun. Number Theory Phys. 72, Faber, C., Pandharipande, R.: Hodge integrals and Gromov Witten theory. Invent. Math. 39, Gessel, I.M.: On Miki s identity for Bernoulli numbers. J. Number Theory, Kim, D.S., Dolgy, D.V., Kim, T., Rim, S.-H.: Identities involving Bernoulli and Euler polynomials arising from Chebyshev polynomials. Proc. Jangjeon Math. Soc. 54, Kim, D.S., Kim, T., Lee, S.-H.: Some identities for Bernoulli polynomials involving Chebyshev polynomials. J. Comput. Anal. Appl. 6, Kim, T., Kim, D.S., Jang, G.-W., Kwon, J.: Sums of finite products of Euler functions. In: Advances in Real and Complex Analysis with Applications. Trends in Mathematics, 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, Kim, T., Kim, D.S., Jang, L.-C., Jang, G.-W.: Fourier series of sums of products of Bernoulli functions and their applications. J. Nonlinear Sci. Appl., Mason, J.C.: Chebyshev polynomials of the second, third and fourth kinds in approximation, indefinite integration, and integral transforms. J. Comput. Appl. Math. 49, Miki, H.: A relation between Bernoulli numbers. J. Number Theory 3, Prodinger, H.: Representing derivatives of Chebyshev polynomials by Chebyshev polynomials and related questions. Open Math. 5, Shiratani, K., Yokoyama, S.: An application of p-adic convolutions. Mem. Fac. Sci., Kyushu Univ., Ser. A, Math. 36, Yuan, Y., Zhang, W.: Some identities involving the Fibonacci polynomials. Fibonacci Q. 4, Zhang, W.: Some identities involving the Fibonacci numbers and Lucas numbers. Fibonacci Q. 42,