Discovering and Proving Infinite Binomial Sums Identities

References

• [Ablinger 14] J. Ablinger. “The Package HarmonicSums: Computer Algebra and Analytic Aspects of Nested Sums.” In Loops and Legs in Quantum Field Theory – LL, edited by J. Bluemlein, P. Marquard, T. Riemann, 2014. http://arxiv.org/abs/1407.6180.
   Google Scholar
• [Ablinger and Blümlein 13] J. Ablinger and J. Blümlein. “Harmonic Sums, Polylogarithms, Special Numbers, and Their Generalizations.” In Computer Algebra in Quantum Field Theory: Integration, Summation and Special Functions, Texts & Monographs in Symbolic Computation, edited by C. Schneider and J. Blümlein, pp. 1–32. Wien: Springer, 2013. http://arxiv.org/abs/1304.7071
   Google Scholar
• [Ablinger et al. 14] J. Ablinger, J. Blümlein, and C. Schneider. “Generalized Harmonic, Cyclotomic, and Binomial Sums, Their Polylogarithms and Special Numbers.” J. Phys. Conf. Ser. 523 (2014). http://arxiv.org/abs/1310.5645.
   Google Scholar
• [Ablinger 12] J. Ablinger. “Computer Algebra Algorithms for Special Functions in Particle Physics.” J. Kepler University Linz. PhD Thesis. April 2012.
   Google Scholar
• [Ablinger et al. 14] J. Ablinger, J. Blümlein, C. G. Raab, and C. Schneider. “Iterated Binomial Sums and Their Associated Iterated Integrals.” J. Math. Phys. Comput. 55 (2014), 1–57. http://arxiv.org/abs/1407.1822.
   Google Scholar
• [Ablinger et al. 11] J. Ablinger, J. Blümlein, and C. Schneider. “Harmonic Sums and Polylogarithms Generated by Cyclotomic Polynomials.” J. Math. Phys. 52(10) (2011), 1–52. http://arxiv.org/abs/1105.6063.
   Web of Science ®Google Scholar
• [Ablinger et al. 13] J. Ablinger, J. Blümlein, and C. Schneider. “Analytic and Algorithmic Aspects of Generalized Harmonic Sums and Polylogarithms.” J. Math. Phys. 54 082301 (2013), 1–74. http://arxiv.org/abs/1302.0378.
   PubMed Web of Science ®Google Scholar
• [Borwein and Borwein 87] J. M. Borwein and P. B. Borwein. Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, 1987. Reprinted 1998.
   Google Scholar
• [Borwein et al. 01] J. M. Borwein, D. J. Broadhurst, and J. Kamnitzer. “Central Binomial Sums, Multiple Clausen Values, and Zeta Values.” Exp. Math. 10 (2001), 25–34. http://arxiv.org/abs/hep-th/0004153.
   Web of Science ®Google Scholar
• [Borwein and Lisoněk 00] J. M. Borwein and P. Lisoněk. “Applications of Integer Relation Algorithms.” Discrete Math. 217 (2000), 65–82.
   Web of Science ®Google Scholar
• [Davydychev and Kalmykov 01] A. I. Davydychev and M. Y. Kalmykov. “New Results for the Epsilon-Expansion of Certain One-, Two- and Three-Loop Feynman Diagrams.” Nucl. Phys. B 605 (2001), 266–318. http://arxiv.org/abs/hep-th/0012189.
   Web of Science ®Google Scholar
• [Davydychev and Kalmykov 04] A. I. Davydychev and M. Y. Kalmykov. “Massive Feynman Diagrams and Inverse Binomial Sums.” Nucl. Phys. B 699 (2004), 3–64. http://arxiv.org/abs/hep-th/0303162.
   Web of Science ®Google Scholar
• [Fleischer et al. 99] J. Fleischer, A. V. Kotikov, and O. L. Veretin. “Analytic Two Loop Results for Selfenergy Type and Vertex Type Diagrams with One Nonzero Mass.” Nucl. Phys. B 547 (1999), 343–374. http://arxiv.org/abs/hep-ph/9808242.
   Web of Science ®Google Scholar
• [Hoffman 00] M. Hoffman. “Quasi-Shuffle Products.” J. Algebraic Combin. 11 (2000), 49–68. http://arxiv.org/abs/math/9907173.
   Web of Science ®Google Scholar
• [Jegerlehner et al. 03] F. Jegerlehner, M. Y. Kalmykov, and O. Veretin. “MS Versus Pole Masses of Gauge Bosons II: Two-Loop Electroweak Fermion Corrections.” Nucl. Phys. B 658 (2003), 49–112. http://arxiv.org/abs/hep-ph/0212319.
   Web of Science ®Google Scholar
• [Kalmykov and Veretin 00] M. Y. Kalmykov and O. Veretin. “Single Scale Diagrams and Multiple Binomial Sums.” Phys. Lett. B 483 (2000), 315–323. http://arxiv.org/abs/hep-th/0004010.
   Web of Science ®Google Scholar
• [Kalmykov et al. 07] M. Y. Kalmykov, B. F. L. Ward, and S. A. Yost. “Multiple (Inverse) Binomial Sums of Arbitrary Weight and Depth and the All-Order ϵ-Expansion of Generalized Hypergeometric Functions with One Half-Integer Value of Parameter.” JHEP 0710 (2007), 048. http://arxiv.org/abs/0707.3654.
   Google Scholar
• [Lehmer 85] D. H. Lehmer. “Interesting Series Involving the Central Binomial Coefficient.” Amer. Math. Mon. 92 (1985), 449–457.
   Web of Science ®Google Scholar
• [Ogreid and Osland 98] O. M. Ogreid and P. Osland. “Summing One-Dimensional and Two-Dimensional Series Related to the Euler Series.” J. Comput. Appl. Math. 98 (1998), 245–271. http://arxiv.org/abs/hep-th/9801168.
   Web of Science ®Google Scholar
• [Remiddi and Vermaseren 00] E. Remiddi and J. A. M. Vermaseren. “Harmonic Polylogarithms.” Int. J. Mod. Phys. A 15 (2000), 725–754. http://arxiv.org/abs/hep-ph/9905237.
   Web of Science ®Google Scholar
• [Sun 14] Zhi-Wei Sun. “List of Conjectural Series for Powers of π and Other Constants.” http://arxiv.org/abs/1102.5649, 2014.
   Google Scholar
• [Weinzierl 04] S. Weinzierl. “Expansion Around Half Integer Values, Binomial Sums and Inverse Binomial Sums.” J. Math. Phys. 45 (2004), 2656–2673. http://arxiv.org/abs/hep-ph/0402131.
   Web of Science ®Google Scholar
• [Zucker 85] I. J. Zucker. “On the Series and Related Sums.” J. Number Theory 20 (1985), 92–102.
   Web of Science ®Google Scholar