Abstract
We employ computer algebra algorithms to prove a collection of identities involving Bessel functions with half-integer orders and other special functions. These identities appear in the famous Handbook of Mathematical Functions, as well as in its successor, the DLMF, but their proofs were lost. We use generating functions and symbolic summation techniques to produce new proofs for them.
In memory of Frank W.J. Olver (1924–2013)
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Notes
- 1.
Finally Chap. 10 Bessel Functions.
References
Ablinger, J., Blümlein, J., Round, M., Schneider, C.: Advanced Computer Algebra Algorithms for the Expansion of Feynman Integrals. In: Loops and Legs in Quantum Field Theory 2012, PoS(2012), Wernigerode, pp. 1–14 (2012)
Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions. Dover, New York (1973). A reprint of the tenth National Bureau of Standards edition, 1964
Andrews, G.E., Paule, P., Schneider, C.: Plane partitions VI: Stembridge’s TSPP theorem. Adv. Appl. Math. 34(4), 709–739 (2005). (Special Issue Dedicated to Dr. David P. Robbins. Edited by D. Bressoud)
Chyzak, F.: An extension of Zeilberger’s fast algorithm to general holonomic functions. Discrete Math. 217, 115–134 (2000)
Gerhold, S.: Uncoupling systems of linear Ore operator equations. Master’s thesis, RISC, J. Kepler University, Linz (2002)
Gerhold, S.: The Hartman-Watson distribution revisited: asymptotics for pricing Asian options. J. Appl. Probab. 48(3), 892–899 (2011)
Graham, R.L., Knuth, D.E., Patashnik, O.: Concrete Mathematics, 2nd edn. Addison Wesley, Reading (1994)
Ince, E.L.: Ordinary Differential Equations. Dover, New York (1926)
Kauers, M.: The holonomic toolkit. In: Blümlein, J., Schneider, C. (eds.) Computer Algebra in Quantum Field Theory: Integration, Summation and Special Functions. Texts and Monographs in Symbolic Computation. Springer, Vienna (2013)
Kauers, M., Paule, P.: The Concrete Tetrahedron. Springer, Wien/New York (2011)
Koutschan, C.: Advanced applications of the holonomic systems approach. PhD thesis, RISC, J. Kepler University, Linz (2009)
Koutschan, C., Moll, V.H.: The integrals in Gradshteyn and Ryzhik. Part 18: some automatic proofs. SCIENTIA Ser. A Math. Sci. 20, 93–111 (2011)
Mallinger, C.: Algorithmic manipulations and transformations of univariate holonomic functions and sequences. Master’s thesis, RISC, J. Kepler University, Linz (1996)
Meunier, L., Salvy, B.: ESF: an automatically generated encyclopedia of special functions. In: Rafael Sendra, J. (ed.) Proceedings of the ISSAC’03, Philadelphia, pp. 199–206. ACM (2003)
Olver, F.W.J., Lozier, D.W., Boisvert, R.F., Clark, C.W. (eds.): NIST digital library of mathematical functions. http://dlmf.nist.gov/. Release 1.0.5 of 2012-10-01
Paule, P., Schorn, M.: A mathematica version of Zeilberger’s algorithm for proving binomial coefficient identities. J. Symb. Comput. 20(5–6), 673–698 (1995)
Prause, K.: The generalized hyperbolic model: estimation, financial derivatives, and risk measures. PhD thesis, Albert-Ludwigs-Universität, Freiburg i. Br. (1999)
Salvy, B., Zimmermann, P.: Gfun: a package for the manipulation of generating and holonomic functions in one variable. ACM Trans. Math. Softw. 20, 163–177 (1994)
Schneider, C.: A new Sigma approach to multi-summation. Adv. Appl. Math. 34(4), 740–767 (2005). (Special Issue Dedicated to Dr. David P. Robbins. Edited by D. Bressoud)
Schneider, C.: Symbolic summation assists combinatorics. Sém. Lothar. Comb. 56, 1–36, (2007). (Article B56b)
Szegö, G.: Orthogonal Polynomials. Volume XXIII of Colloquium Publications, 4th edn. AMS, Providence (1975)
Zeilberger, D.: The method of creative telescoping. J. Symb. Comput. 11, 195–204 (1991)
Acknowledgements
This work has been supported by the Austrian Science Fund (FWF) grants P20347-N18, P24880-N25, Y464-N18, DK W1214 (DK6, DK13) and SFB F50 (F5004-N15, F5006-N15, F5009-N15), and by the EU Network LHCPhenoNet PITN-GA-2010-264564.
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Gerhold, S., Kauers, M., Koutschan, C., Paule, P., Schneider, C., Zimmermann, B. (2013). Computer-Assisted Proofs of Some Identities for Bessel Functions of Fractional Order. In: Schneider, C., Blümlein, J. (eds) Computer Algebra in Quantum Field Theory. Texts & Monographs in Symbolic Computation. Springer, Vienna. https://doi.org/10.1007/978-3-7091-1616-6_3
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