A new type of Taylor series expansion
© The Author(s) 2018
Received: 29 January 2018
Accepted: 4 May 2018
Published: 15 May 2018
We present a variant of the classical integration by parts to introduce a new type of Taylor series expansion and to present some closed forms for integrals involving Jacobi and Laguerre polynomials, which cannot be directly obtained by usual symbolic computation programs, i.e., only some very specific values can be computed by the mentioned programs. An error analysis is given in the sequel for the introduced expansion.
Davis also mentioned that the expansion of a function based on a series of predetermined (basis) functions can be interpreted as an interpolation problem with an infinite number of conditions. See also  in this regard.
The problem of the representation of an arbitrary function by means of linear combinations of prescribed functions has received a lot of attention in approximation theory. It is well known that a special case of this problem directly leads to the Taylor series expansion where the prescribed functions are monomial bases .
The main aim of this paper is to introduce a new type of Taylor series expansion through a variant of the classical integration by parts. In the next section, we present the general form of this expansion and consider some interesting cases of it leading to new closed forms for integrals involving Jacobi and Laguerre polynomials. Also, an error analysis is given in Sect. 3 for the introduced expansion.
2 A new type of Taylor series expansion
Let F and G be two smooth enough functions such that repeated differentiation and repeated integration by parts are allowed for them. The rule of integration by parts  allows one to perform successive integrations on the integrals of the form \(\int F(t) G(t) \,dt\) without tedious algebraic computations.
Formula (1) provides a straightforward proof for Taylor’s theorem with an integral remainder term, according to the following result.
Now, let us consider some particular examples of the main formula (3). We would like to notice here that the closed forms for the integrals involving Jacobi and Laguerre polynomials in the following examples are new in the literature (see, e.g., [8, 9]) to the best of our knowledge, and they can be computed only for specific values of the parameters by using symbolic computation.
For more certain new, interesting, and useful integrals and expansion formulas involving the hypergeometric function and the Laguerre polynomials, see .
3 Error analysis
The work of the first author has been supported by the Alexander von Humboldt Foundation under the grant number: Ref 3.4–IRN–1128637–GF-E. The third and fourth authors thank for the financial support from the Agencia Estatal de Innovación (AEI) of Spain under grant MTM2016–75140–P, co-financed by the European Community fund FEDER and Xunta de Galicia, grants GRC 2015–004 and R 2016/022.
Each of the authors contributed to each part of this study equally and read and approved the final version of the paper.
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
- Mastroianni, G., Milovanović, G.V.: Interpolation Processes—Basic Theory and Applications. Springer Monographs in Mathematics. Springer, Berlin (2008) View ArticleMATHGoogle Scholar
- Sloan, I.H.: Nonpolynomial interpolation. J. Approx. Theory 39, 97–117 (1983) MathSciNetView ArticleMATHGoogle Scholar
- Davis, P.J.: Interpolation and Approximation. Dover, New York (1975) MATHGoogle Scholar
- Masjed-Jamei, M.: On constructing new interpolation formulas using linear operators and an operator type of quadrature rules. J. Comput. Appl. Math. 216, 307–318 (2008) MathSciNetView ArticleMATHGoogle Scholar
- Masjed-Jamei, M.: On constructing new expansions of functions using linear operators. J. Comput. Appl. Math. 234, 365–374 (2010) MathSciNetView ArticleMATHGoogle Scholar
- Widder, D.V.: A generalization of Taylor’s series. Trans. Am. Math. Soc. 30, 126–154 (1927) MathSciNetMATHGoogle Scholar
- Horowitz, D.: Tabular integration by parts. Coll. Math. J. 21, 307–311 (1990) View ArticleGoogle Scholar
- Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series and Products. Academic Press, London (1980) MATHGoogle Scholar
- Prudnikov, A.P., Brychkov, Yu.A., Marichev, O.I.: Integrals and Series, Volumes 2 and 3. Gordon & Breach, New York (1986) MATHGoogle Scholar
- Szegö, G.: Orthogonal Polynomials. Am. Math. Soc., Providence (1975) MATHGoogle Scholar
- Agarwal, P., Qi, F., Chand, M., Jain, S.: Certain integrals involving the generalized hypergeometric function and the Laguerre polynomials. J. Comput. Appl. Math. 313, 307–317 (2017) MathSciNetView ArticleMATHGoogle Scholar
- Slater, L.J.: Generalized Hypergeometric Functions. Cambridge University Press, Cambridge (1966) MATHGoogle Scholar
- Agarwal, P., Jain, S., Choi, J.: Certain q-series identities. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 111, 139–146 (2017) MathSciNetView ArticleMATHGoogle Scholar
- Ruzhansky, M., Cho, Y.J., Agarwal, P., Area, I. (eds.): Advances in Real and Complex Analysis with Applications. Springer, Singapore (2017) Google Scholar