A refined difference field theory for symbolic summation. (English) Zbl 1147.33006
The author introduces the basic summation problems in difference fields and presents in summarized form a refined summation theory of depth-optimal \(\Pi\Sigma^*\)-extensions in which the central results are supplemented by concrete examples.
Applications from particle physics are presented.
Applications from particle physics are presented.
Reviewer: Hans Benker (Merseburg)
MSC:
| 33F10 | Symbolic computation of special functions (Gosper and Zeilberger algorithms, etc.) |
| 68W30 | Symbolic computation and algebraic computation |
References:
| [1] | Abramov, S.; Petkovšek, M., D’Alembertian solutions of linear differential and difference equations, (von zur Gathen, J., Proc. ISSAC’94 (1994), ACM Press), 169-174 · Zbl 0919.34013 |
| [2] | Andrews, G.; Askey, R.; Roy, R., Special Functions (2006), Cambridge University Press |
| [3] | Bauer, A.; Petkovšek, M., Multibasic and mixed hypergeometric Gosper-type algorithms, J. Symbolic Comput., 28, 4-5, 711-736 (1999) · Zbl 0973.33010 |
| [4] | Bierenbaum, I., Blümlein, J., Klein, S., Schneider, C., 2007. Difference equations in massive higher order calculations, in: Proc. ACAT 2007, volume PoS(ACAT)082; Bierenbaum, I., Blümlein, J., Klein, S., Schneider, C., 2007. Difference equations in massive higher order calculations, in: Proc. ACAT 2007, volume PoS(ACAT)082 |
| [5] | Blümlein, J., Algebraic relations between harmonic sums and associated quantities, Comput. Phys. Commun., 159, 4, 19-54 (2004) · Zbl 1097.11063 |
| [6] | Blümlein, J.; Kurth, S., Harmonic sums and Mellin transforms up to two-loop order, Phys. Rev., D60 (1999) |
| [7] | Bronstein, M., On solutions of linear ordinary difference equations in their coefficient field, J. Symbolic Comput., 29, 6, 841-877 (2000) · Zbl 0961.12004 |
| [8] | Chyzak, F., An extension of Zeilberger’s fast algorithm to general holonomic functions, Discrete Math., 217, 115-134 (2000) · Zbl 0968.33011 |
| [9] | Driver, K.; Prodinger, H.; Schneider, C.; Weideman, J., Padé approximations to the logarithm III: Alternative methods and additional results, Ramanujan J., 12, 3, 299-314 (2006) · Zbl 1108.41011 |
| [10] | Gosper, R., Decision procedures for indefinite hypergeometric summation, Proc. Natl. Acad. Sci. USA, 75, 40-42 (1978) · Zbl 0384.40001 |
| [11] | Hendriks, P.; Singer, M., Solving difference equations in finite terms, J. Symbolic Comput., 27, 3, 239-259 (1999) · Zbl 0930.39004 |
| [12] | Karr, M., Summation in finite terms, J. ACM, 28, 305-350 (1981) · Zbl 0494.68044 |
| [13] | Karr, M., Theory of summation in finite terms, J. Symbolic Comput., 1, 303-315 (1985) · Zbl 0585.68052 |
| [14] | Kauers, M.; Schneider, C., Application of unspecified sequences in symbolic summation, (Dumas, J., Proc. ISSAC’06 (2006), ACM Press), 177-183 · Zbl 1356.68284 |
| [15] | Kauers, M.; Schneider, C., Indefinite summation with unspecified summands, Discrete Math., 306, 17, 2021-2140 (2006) · Zbl 1157.05006 |
| [16] | Kauers, M.; Schneider, C., Symbolic summation with radical expressions, (Brown, C., Proc. ISSAC’07 (2007)), 219-226 · Zbl 1190.68091 |
| [17] | Kuba, M., Prodinger, H., Schneider, C., 2005. Generalized reciprocity laws for sums of harmonic numbers. SFB-Report 2005-17, J. Kepler University Linz; Kuba, M., Prodinger, H., Schneider, C., 2005. Generalized reciprocity laws for sums of harmonic numbers. SFB-Report 2005-17, J. Kepler University Linz · Zbl 1202.68492 |
| [18] | Moch, S., Schneider, C., 2007. Feynman integrals and difference equations, in: Proc. ACAT 2007, volume PoS(ACAT)083; Moch, S., Schneider, C., 2007. Feynman integrals and difference equations, in: Proc. ACAT 2007, volume PoS(ACAT)083 |
| [19] | Osburn, R., Schneider, C., Gaussian hypergeometric series and extensions of supercongruences. Math. Comp., 2008 (in press); Osburn, R., Schneider, C., Gaussian hypergeometric series and extensions of supercongruences. Math. Comp., 2008 (in press) · Zbl 1209.11049 |
| [20] | Paule, P., Greatest factorial factorization and symbolic summation, J. Symbolic Comput., 20, 3, 235-268 (1995) · Zbl 0854.68047 |
| [21] | Paule, P.; Riese, A., A Mathematica q-analogue of Zeilberger’s algorithm based on an algebraically motivated aproach to \(q\)-hypergeometric telescoping, (Ismail, M.; Rahman, M., Special Functions, q-Series and Related Topics, vol. 14 (1997), Fields Institute Toronto, AMS), 179-210 · Zbl 0869.33010 |
| [22] | Paule, P.; Schneider, C., Computer proofs of a new family of harmonic number identities, Adv. in Appl. Math., 31, 2, 359-378 (2003) · Zbl 1039.11007 |
| [23] | Paule, P.; Schneider, C., Truncating binomial series with symbolic summation, INTEGERS. Electron. J. Combin. Number Theory, 7, 1-9 (2007), #A22 · Zbl 1120.05005 |
| [24] | Pemantle, R.; Schneider, C., When is 0.999... equal to 1?, Amer. Math. Monthly, 114, 4, 344-350 (2007) · Zbl 1226.11134 |
| [25] | Risch, R., The problem of integration in finite terms, Trans. Amer. Math. Soc., 139, 167-189 (1969) · Zbl 0184.06702 |
| [26] | Risch, R., The solution to the problem of integration in finite terms, Bull. Amer. Math. Soc., 76, 605-608 (1970) · Zbl 0196.06801 |
| [27] | Schneider, C., 2001. Symbolic summation in difference fields. Technical Report 01-17, RISC-Linz, J. Kepler University, Ph.D. Thesis; Schneider, C., 2001. Symbolic summation in difference fields. Technical Report 01-17, RISC-Linz, J. Kepler University, Ph.D. Thesis |
| [28] | Schneider, C., The summation package Sigma: Underlying principles and a rhombus tiling application, Discrete Math. Theor. Comput. Sci., 6, 2, 365-386 (2004) · Zbl 1066.68164 |
| [29] | Schneider, C., Symbolic summation with single-nested sum extensions, (Gutierrez, J., Proc. ISSAC’04 (2004), ACM Press), 282-289 · Zbl 1134.33329 |
| [30] | Schneider, C., Finding telescopers with minimal depth for indefinite nested sum and product expressions, (Kauers, M., Proc. ISSAC’05 (2005), ACM), 285-292 · Zbl 1360.68953 |
| [31] | Schneider, C., A new Sigma approach to multi-summation, Adv. in Appl. Math. Special Issue Dedicated to Dr. David P. Robbins, 34, 4, 740-767 (2005) · Zbl 1078.33021 |
| [32] | Schneider, C., Product representations in \(\Pi \Sigma \)-fields, Ann. Comb., 9, 1, 75-99 (2005) · Zbl 1123.33021 |
| [33] | Schneider, C., Solving parameterized linear difference equations in terms of indefinite nested sums and products, J. Difference Equ. Appl., 11, 9, 799-821 (2005) · Zbl 1087.33011 |
| [34] | Schneider, C., Simplifying Sums in \(\Pi \Sigma \)-Extensions, J. Algebra Appl., 6, 3, 415-441 (2007) · Zbl 1120.33023 |
| [35] | Schneider, C., Symbolic summation assists combinatorics, Sém. Lothar. Combin., 56, 1-36 (2007), Article B56b · Zbl 1188.05001 |
| [36] | Schneider, C., 2007c. Symbolic summation finds optimal nested sum representations. SFB-Report 2007-26, SFB F013, J. Kepler University Linz; Schneider, C., 2007c. Symbolic summation finds optimal nested sum representations. SFB-Report 2007-26, SFB F013, J. Kepler University Linz |
| [37] | Schneider, C., 2008. Parameterized telescoping proves algebraic independence of sums. Ann. Comb. (in press); Schneider, C., 2008. Parameterized telescoping proves algebraic independence of sums. Ann. Comb. (in press) · Zbl 1232.33034 |
| [38] | Singer, M.; Saunders, B.; Caviness, B., An extension of Liouville’s theorem on integration in finite terms, SIAM J. Comput., 14, 4, 966-990 (1985) · Zbl 0575.12021 |
| [39] | Vermaseren, J., Harmonic sums, Mellin transforms and integrals, Internat. J. Modern Phys., A14, 2037-2976 (1999) · Zbl 0939.65032 |
| [40] | Vermaseren, J.; Vogt, A.; Moch, S., The third-order qcd corrections to deep-inelastic scattering by photon exchange, Nuclear Phys., B724, 3-182 (2005) · Zbl 1178.81286 |
| [41] | Zeilberger, D., The method of creative telescoping, J. Symbolic Comput., 11, 195-204 (1991) · Zbl 0738.33002 |
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