Derivatives of any Horn-type hypergeometric functions with respect to their parameters. (English) Zbl 1481.33012
Summary: We consider the derivatives of Horn hypergeometric functions of any number of variables with respect to their parameters. The derivative of such a function of \(n\) variables is expressed as a Horn hypergeometric series of \(n+1\) infinite summations depending on the same variables and with the same region of convergence as for the original Horn hypergeometric function. The derivatives of Appell functions, generalized hypergeometric functions, confluent and non-confluent Lauricella series, and generalized Lauricella series are explicitly presented. Applications to the calculations of Feynman diagrams are discussed, especially the series expansions in \(\epsilon\) within dimensional regularization. Connections with other classes of special functions are discussed as well.
MSC:
| 33C65 | Appell, Horn and Lauricella functions |
References:
| [1] | Miller, A. R., Summations for certain series containing the digamma function, J. Phys. A, Math. Gen., 39, 3011 (2006) · Zbl 1091.33004 |
| [2] | Cvijović, D.; Miller, A. R., A reduction formula for the Kampé de Fériet function, Appl. Math. Lett., 23, 769 (2010) · Zbl 1190.33008 |
| [3] | Cvijović, D., Closed-form summations of certain hypergeometric-type series containing the digamma function, J. Phys. A, Math. Theor., 41, Article 455205 pp. (2008) · Zbl 1156.33001 |
| [4] | Ancarani, L. U.; Gasaneo, G., Derivatives of any order of the Gaussian hypergeometric function \({}_2F_1(a, b, c; z)\) with respect to the parameters a, b and c, J. Phys. A, Math. Theor., 42, Article 395208 pp. (2009) · Zbl 1183.33005 |
| [5] | Ancarani, L. U.; Gasaneo, G., Derivatives of any order of the hypergeometric function \({}_pF_q( a_1, . . ., a_p; b_1, . ., b_q; z)\) with respect to the parameters \(a_i\) and \(b_i\), J. Phys. A, Math. Theor., 43, Article 085210 pp. (2010) · Zbl 1190.33007 |
| [6] | Ancarani, L. U.; Del Punta, J. A.; Gasaneo, G., Derivatives of Horn hypergeometric functions with respect to their parameters, J. Math. Phys., 58, Article 073504 pp. (2017) · Zbl 1370.33015 |
| [7] | Sahai, V.; Verma, A., Derivatives of Appell functions with respect to parameters, J. Inequal. Spec. Funct., 6, 3, 1 (2015) |
| [8] | Fejzullahu, B. Xh., Parameter derivatives of the generalized hypergeometric function, Integral Transforms Spec. Funct., 28, 781 (2017) · Zbl 1379.33011 |
| [9] | Greynat, D.; Sesma, J., A new approach to the epsilon expansion of generalized hypergeometric functions, Comput. Phys. Commun., 185, 472 (2014) · Zbl 1348.33005 |
| [10] | Greynat, D.; Sesma, J.; Vulvert, G., Derivatives of the Pochhammer and reciprocal Pochhammer symbols and their use in epsilon-expansions of Appell and Kampé de Fériet functions, J. Math. Phys., 55, Article 043501 pp. (2014) · Zbl 1292.81102 |
| [11] | Hansen, E. R., A Table of Series and Products, Prentice-Hall Series in Automatic Computation, vol. XVIII (1975), Prentice-Hall Inc.: Prentice-Hall Inc. Englewood Cliffs, N.J. · Zbl 0438.00001 |
| [12] | Abramowitz, M.; Stegun, I. A., Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables (1974), Dover Publications Inc.: Dover Publications Inc. New York, N.Y. · Zbl 0515.33001 |
| [13] | Brychkov, Yu. A., Handbook of Special Functions: Derivatives, Integrals, Series and Other Formulas (2008), Chapman and Hall/CRC Press: Chapman and Hall/CRC Press New York · Zbl 1158.33001 |
| [14] | Fröhlich, J., Parameter derivatives of the jacoby polynomials and the gaussian hypergeometric function, Integral Transforms Spec. Funct., 2, 253 (1994) · Zbl 0822.33003 |
| [15] | Moch, S.; Uwer, P.; Weinzierl, S., Nested sums, expansion of transcendental functions, and multiscale multiloop integrals, J. Math. Phys., 43, 3363 (2002) · Zbl 1060.33007 |
| [16] | Srivastava, H. M.; Manocha, H. L., A Treatise on Generating Functions, Ellis Horwood Series in Mathematics and Its Applications (1984), Ellis Horwood Ltd.: Ellis Horwood Ltd. Chichester · Zbl 0535.33001 |
| [17] | Srivastava, H. M.; Karlsson, P. W., Multiple Gaussian Hypergeometric Series, Ellis Horwood Series in Mathematics and Its Applications (1985), Ellis Horwood Ltd.: Ellis Horwood Ltd. Chichester · Zbl 0552.33001 |
| [18] | de Calan, C.; Malbouisson, A. P.C., Complete Mellin representation and asymptotic behaviours of Feynman amplitudes, Ann. Inst. Henri Poincaré. Phys. Théor., 32, 91 (1980) |
| [19] | de Calan, C.; David, F.; Rivasseau, V., Renormalization in the complete Mellin representation of Feynman amplitudes, Commun. Math. Phys., 78, 531 (1981) |
| [20] | de Calan, C.; Malbouisson, A. P.C., Infrared and ultraviolet dimensional meromorphy of Feynman amplitudes, Commun. Math. Phys., 90, 413 (1983) |
| [21] | Smirnov, V. A., Analytic Tools for Feynman Integrals, Springer Tracts Mod. Phys., vol. 250, 1 (2012) · Zbl 1268.81004 |
| [22] | Bytev, V. V.; Kalmykov, M. Yu.; Kniehl, B. A., Differential reduction of generalized hypergeometric functions from Feynman diagrams: One-variable case, Nucl. Phys. B, 836, 129 (2010) · Zbl 1206.81089 |
| [23] | Davydychev, A. I., Some exact results for N-point massive Feynman integrals, J. Math. Phys., 32, 1052 (1991) · Zbl 0729.58054 |
| [24] | Davydychev, A. I., General results for massive N-point Feynman diagrams with different masses, J. Math. Phys., 33, 358 (1992) |
| [25] | Fleischer, J.; Jegerlehner, F.; Tarasov, O. V., A new hypergeometric representation of one-loop scalar integrals in d dimensions, Nucl. Phys. B, 672, 303 (2003) · Zbl 1058.81605 |
| [26] | Kniehl, B. A.; Tarasov, O. V., Functional equations for one-loop master integrals for heavy-quark production and Bhabha scattering, Nucl. Phys. B, 820, 178 (2009) · Zbl 1194.81266 |
| [27] | Kniehl, B. A.; Tarasov, O. V., Analytic result for the one-loop scalar pentagon integral with massless propagators, Nucl. Phys. B, 833, 298 (2010) · Zbl 1204.81075 |
| [28] | Jegerlehner, F.; Kalmykov, M. Yu.; Veretin, O., \( \overline{\operatorname{MS}}\) vs. pole masses of gauge bosons II: two-loop electroweak fermion corrections, Nucl. Phys. B, 658, 49 (2003) · Zbl 1097.81929 |
| [29] | Davydychev, A. I.; Kalmykov, M. Yu., Massive Feynman diagrams and inverse binomial sums, Nucl. Phys. B, 699, 3 (2004) · Zbl 1123.81388 |
| [30] | Kalmykov, M. Yu., Gauss hypergeometric function: reduction, ϵ-expansion for integer/half-integer parameters and Feynman diagrams, J. High Energy Phys., 0604, Article 056 pp. (2006) |
| [31] | Kalmykov, M. Yu.; Ward, B. F.L.; Yost, S., All order ϵ-expansion of Gauss hypergeometric functions with integer and half/integer values of parameters, J. High Energy Phys., 0702, Article 040 pp. (2007) |
| [32] | Kalmykov, M. Y.; Ward, B. F.L.; Yost, S. A., On the all-order ϵ-expansion of generalized hypergeometric functions with integer values of parameters, J. High Energy Phys., 0711, Article 009 pp. (2007) · Zbl 1245.33007 |
| [33] | Kalmykov, M. Yu.; Kniehl, B. A., All-order ϵ expansions of hypergeometric functions of one variable, Phys. Part. Nucl., 41, 942 (2010) |
| [34] | Yost, S. A.; Bytev, V. V.; Kalmykov, M. Yu.; Kniehl, B. A.; Ward, B. F.L., The Epsilon Expansion of Feynman Diagrams via Hypergeometric Functions and Differential Reduction |
| [35] | Weinzierl, S., Symbolic expansion of transcendental functions, Comput. Phys. Commun., 145, 357 (2002) · Zbl 1001.65025 |
| [36] | Weinzierl, S., Expansion around half integer values, binomial sums and inverse binomial sums, J. Math. Phys., 45, 2656 (2004) · Zbl 1071.33018 |
| [37] | Moch, S.; Uwer, P., -XSummer- Transcendental functions and symbolic summation in FORM, Comput. Phys. Commun., 174, 759 (2006) · Zbl 1196.68332 |
| [38] | Huber, T.; Maître, D., HypExp, a Mathematica package for expanding hypergeometric functions around integer-valued parameters, Comput. Phys. Commun., 175, 122 (2006) · Zbl 1196.68326 |
| [39] | Huber, T.; Maître, D., HypExp 2, Expanding hypergeometric functions about half-integer parameters, Comput. Phys. Commun., 178, 755 (2008) · Zbl 1196.81024 |
| [40] | Huang, Z.-W.; Liu, J., NumExp: Numerical epsilon expansion of hypergeometric functions, Comput. Phys. Commun., 184, 1973 (2013) · Zbl 1344.33001 |
| [41] | Blümlein, J.; Schneider, C., Analytic computing methods for precision calculations in quantum field theory, Int. J. Mod. Phys. A, 33, Article 1830015 pp. (2018) · Zbl 1392.81192 |
| [42] | Bytev, V. V.; Kalmykov, M. Yu.; Kniehl, B. A., , HYPERgeometric functions DIfferential REduction: MATHEMATICA-based packages for differential reduction of generalized hypergeometric functions \({}_pF_{p - 1}, F_1, F_2, F_3, F_4\), Comput. Phys. Commun., 184, 2332 (2013) · Zbl 07866097 |
| [43] | Bytev, V. V.; Kniehl, B. A., HYPERDIRE HYPERgeometric functions DIfferential REduction: Mathematica-based packages for the differential reduction of generalized hypergeometric functions: Horn-type hypergeometric functions of two variables, Comput. Phys. Commun., 189, 128 (2015) · Zbl 1344.33006 |
| [44] | Bytev, V. V.; Kalmykov, M. Yu.; Moch, S.-O., HYPERgeometric functions DIfferential REduction (): MATHEMATICA based packages for differential reduction of generalized hypergeometric functions: \( F_D\) and \(F_S\) Horn-type hypergeometric functions of three variables, Comput. Phys. Commun., 185, 3041 (2014) · Zbl 1348.33001 |
| [45] | Bytev, V. V.; Kniehl, B. A., HYPERDIRE—HYPERgeometric functions DIfferential REduction: Mathematica-based packages for the differential reduction of generalized hypergeometric functions: Lauricella function \(F_C\) of three variables, Comput. Phys. Commun., 206, 78 (2016) · Zbl 1375.33002 |
| [46] | Kniehl, B. A.; Kotikov, A. V.; Onishchenko, A. I.; Veretin, O. L., Two-loop sunset diagrams with three massive lines, Nucl. Phys. B, 738, 306 (2006) · Zbl 1109.81331 |
| [47] | Kniehl, B. A.; Kotikov, A. V.; Onishchenko, A. I.; Veretin, O. L., Two-loop diagrams in nonrelativistic QCD with elliptics, Nucl. Phys. B, 948, Article 114780 pp. (2019) · Zbl 1435.81240 |
| [48] | Appell, P., Sur les séries hypergéométriques de deux variables et sur des équations différentielles linéaires aux dérivées partielles, C. R. Hebd. Séances Acad. Sci., 90, 296 (1880), 731 · JFM 12.0296.01 |
| [49] | Appell, P.; Kampé de Fériet, J., Fonctions hypergéométriques et hypersphériques. Polynomes d’Hermite (1926), Gauthier-Villars: Gauthier-Villars Paris · JFM 52.0361.13 |
| [50] | Schlosser, M. J., Multiple Hypergeometric Series: Appell Series and Beyond, (Schneider, C.; Blümlein, J., Computer Algebra in Quantum Field Theory. Computer Algebra in Quantum Field Theory, Texts & Monographs in Symbolic Computation (2013), Springer: Springer Vienna), 305 · Zbl 1310.33013 |
| [51] | Srivastava, H. M., An infinite summation formula associated with Appell’s function \(F_2\), Math. Proc. Camb. Philos. Soc., 65, 679 (1969) · Zbl 0176.01801 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
