In this paper, we use elementary method and some sort of a counting argument to show the equality of two expressions. That is, let $ f(n) $ and $ g(n) $ be two functions, $ k $ be any positive integer. Then $ f(n) = \sum\limits_{r = 0}^n(-1)^r\cdot \frac{n!}{r!}\cdot \binom{n+k-1}{r+k-1}\cdot g(r) $ if and only if $ g(n) = \sum\limits_{r = 0}^n(-1)^r\cdot \frac{n!}{r!}\cdot \binom{n+k-1}{r+k-1}\cdot f(r) $ for all integers $ n\geq0 $. As an application of this formula, we obtain some new identities involving the famous Laguerre polynomials.
Citation: Xiaowei Pan, Xiaoyan Guo. Some new identities involving Laguerre polynomials[J]. AIMS Mathematics, 2021, 6(11): 12713-12717. doi: 10.3934/math.2021733
In this paper, we use elementary method and some sort of a counting argument to show the equality of two expressions. That is, let $ f(n) $ and $ g(n) $ be two functions, $ k $ be any positive integer. Then $ f(n) = \sum\limits_{r = 0}^n(-1)^r\cdot \frac{n!}{r!}\cdot \binom{n+k-1}{r+k-1}\cdot g(r) $ if and only if $ g(n) = \sum\limits_{r = 0}^n(-1)^r\cdot \frac{n!}{r!}\cdot \binom{n+k-1}{r+k-1}\cdot f(r) $ for all integers $ n\geq0 $. As an application of this formula, we obtain some new identities involving the famous Laguerre polynomials.
| [1] | T. M. Apostol, Introduction to analytic number theory, New York: Springer-Verlag, 1976. |
| [2] | T. M. Apostol, Mathematical analysis, 2 Eds., Mass.-London-Don Mills: Addison-Wesley Publishing Co., 1974. |
| [3] | M. Abramowitz, I. A. Stegun, Handbook of mathematical functions, New York, 1965. |
| [4] | D. Jackson, Fourier series and orthogonal polynomials, Dover Publications, 2004. |
| [5] | R. Ma, W. P. Zhang, Several identities involving the Fibonacci numbers and Lucas numbers, Fibonacci Quart., 45 (2007), 164–170. |
| [6] | Y. Yi, W. P. Zhang, Some identities involving the Fibonacci polynomials, Fibonacci Quart., 40 (2002), 314–318. |
| [7] | T. T. Wang, W. P. Zhang, Some identities involving Fibonacci, Lucas polynomials and their applications, Bull. Math. Soc. Sci. Math. Roumanie, 103 (2012), 95–103. |
| [8] | X. H. Wang, D. Han, Some identities related to Dedekind sums and the Chebyshev polynomials, Int. J. Appl. Math. Stat., 51 (2013), 334–339. |
Xiaowei Pan, Xiaoyan Guo. Some new identities involving Laguerre polynomials[J]. AIMS Mathematics, 2021, 6(11): 12713-12717. doi: 10.3934/math.2021733
DownLoad: