×

zbMATH — the first resource for mathematics

The binomial transform of p-recursive sequences and the dilogarithm function. (English) Zbl 07288666
Summary: Using a generalized binomial transform and a novel binomial coefficient identity, we will show that the set of p-recursive sequences is closed under the binomial transform. Using these results, we will derive a new series representation for the dilogarithm function that converges on its domain of analyticity. Finally, we will show that this series representation results in a scheme for numerical evaluation of the dilogarithm function that is accurate, efficient, and stable.
MSC:
11B Sequences and sets
33B99 Elementary classical functions
33F05 Numerical approximation and evaluation of special functions
Software:
DLMF; Maxima; mctoolbox
PDF BibTeX XML Cite
Full Text: Link
References:
[1] Boyadzhiev, K. (2014). Binomial transform and the backward difference, Advances and Applications in Discrete Mathematics, Vol. 18, No. 10, pp. 43-63. · Zbl 1298.05015
[2] Boyadzhiev, K. (2017). Binomial transform of products, Ars Combinatoria, Vol. 126, No. 1, pp. 415-434. · Zbl 1413.11041
[3] AAM: Intern. J., Vol. 15, Issue 2 (December 2020)1031
[4] Boyadzhiev, K. (2018).Notes on the Binomial Transform: Theory and Table with Appendix on Stirling Transform, World Scientific, Singapore. · Zbl 1432.11001
[5] Higham, N. J. (2002).Accuracy and Stability of Numerical Algorithms(second edition), Society for Industrial and Applied Mathematics, Philadelphia, PA, USA. · Zbl 1011.65010
[6] Kauers, M. (2011). The concrete tetrahedron,Proceedings of the 36th International Symposium on Symbolic and Algebraic Computation, ISSAC ’11, pp. 7-8, ACM, New York, NY, USA.
[7] Knuth, D. (1973).The Art of Computer Programming, Vol. 3, Addison-Wesley, Reading, MA. · Zbl 0302.68010
[8] Maxima, a Computer Algebra System (2019). http://maxima.sourceforge.net/
[9] Olver, F., Olde Daalhuis, A., Lozier, D., Schneider, B., Boisvert, R., Clark, C., Miller, B., Saunders, B., Cohl, H., and McClain, M. (2012).NIST Digital Library of Mathematical Functions. http://dlmf.nist.gov/
[10] Prodinger, H. (1994). Some information about the binomial transform, Fibonacci Quarterly, Vol. 32, No. 5, pp. 412-415. · Zbl 0818.05002
[11] Schmidt, M. (2016). Zeta series generating function transformations related to polylogarithm functions and thek-order harmonic numbers, Online Journal of Analytic Combinatorics, Vol. 12, pp. 1-22.
[12] Schneider, C. and Blümlein, J. (2013).Computer Algebra in Quantum Field Theory: Integration, Summation and Special Functions, Springer Publishing Company, Incorporated. · Zbl 1276.81004
[13] Yamada, H. (2015). Generalized binomial transform applied to the divergent series, Acta Physica Polonica B, Vol. 47, No. 11, pp. 2413-2444. · Zbl 1371.44003
[14] Willis, B.L. (2016). Analytic continuation of the3F2hypergeometric series, Integral Transforms and Special Functions, Vol. 21, No. 11, pp. 930-936. · Zbl 1357.33011
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.