On rational and hypergeometric solutions of linear ordinary difference equations in ΠΣ⁎-field extensions

Ablinger, 2018, The two-mass contribution to the three-loop gluonic operator matrix element agg,Q(3), Nucl. Phys. B, 932, 129, 10.1016/j.nuclphysb.2018.04.023 Abramov, 1971, On the summation of rational functions, USSR Comput. Math. Math. Phys., 11, 324, 10.1016/0041-5553(71)90028-0 Abramov, 1989, Problems in computer algebra that are connected with a search for polynomial solutions of linear differential and difference equations, Moscow Univ. Comput. Math. Cybernet., 3, 63 Abramov, 1989, Rational solutions of linear differential and difference equations with polynomial coefficients, USSR Comput. Math. Math. Phys., 29, 7, 10.1016/S0041-5553(89)80002-3 Abramov, 1995, Rational solutions of linear difference and q-difference equations with polynomial coefficients, 285 Abramov, 2006, In memory of Manuel Bronstein, Program. Comput. Softw., 32, 56, 10.1134/S0361768806010063 Abramov, 2000, Hypergeometric dispersion and the orbit problem Abramov, 1994, D'Alembertian solutions of linear differential and difference equations, 169 Abramov, 2010, Polynomial ring automorphisms, rational (w,σ)-canonical forms, and the assignment problem, J. Symb. Comput., 45, 684, 10.1016/j.jsc.2010.03.001 Abramov, 1996, D'Alembertian solutions of inhomogeneous linear equations (differential, difference, and some other), 232 Abramov, 1998, q-hypergeometric solutions of q-difference equations, Discrete Math., 180, 3, 10.1016/S0012-365X(97)00106-4 Abramov, 2020 Bauer, 1999, Multibasic and mixed hypergeometric Gosper-type algorithms, J. Symb. Comput., 28, 711, 10.1006/jsco.1999.0321 Blümlein, 2018, Refined holonomic summation algorithms in particle physics, vol. 226, 51 Bronstein, 1997 Bronstein, 2000, On solutions of linear ordinary difference equations in their coefficient field, J. Symb. Comput., 29, 841, 10.1006/jsco.2000.0368 Bronstein, 1996, An introduction to pseudo-linear algebra, Theor. Comput. Sci., 157, 3, 10.1016/0304-3975(95)00173-5 Chyzak, 2000, An extension of Zeilberger's fast algorithm to general holonomic functions, Discrete Math., 217, 115, 10.1016/S0012-365X(99)00259-9 Chyzak, 1998, Non-commutative elimination in Ore algebras proves multivariate identities, J. Symb. Comput., 26, 187, 10.1006/jsco.1998.0207 Cohn, 1965 Ge, 1993, Testing equalities of multiplicative representations in polynomial time, 422 Hendriks, 1999, Solving difference equations in finite terms, J. Symb. Comput., 27, 239, 10.1006/jsco.1998.0251 Horn, 2012, m-Fold hypergeometric solutions of linear recurrence equations revisited, Math. Comput. Sci., 6, 61, 10.1007/s11786-012-0107-8 Imamoglu, 2017, Computing hypergeometric solutions of second order linear differential equations using quotients of formal solutions and integral bases, J. Symb. Comput., 83, 254, 10.1016/j.jsc.2016.11.014 Karr, 1981, Summation in finite terms, J. ACM, 28, 305, 10.1145/322248.322255 Karr, 1985, Theory of summation in finite terms, J. Symb. Comput., 1, 303, 10.1016/S0747-7171(85)80038-9 Kauers, 2006, Indefinite summation with unspecified summands, Discrete Math., 306, 2073, 10.1016/j.disc.2006.04.005 Kauers, 2007, Symbolic summation with radical expressions, 219 Koutschan, 2013, Creative telescoping for holonomic functions, 171 Middeke, 2018, Denominator bounds for systems of recurrence equations using ΠΣ-extensions, vol. 226, 149 Ocansey, 2018, Representing (q-)hypergeometric products and mixed versions in difference rings, vol. 226, 175 Ore, 1933, Theory of non-commutative polynomials, Ann. Math., 34, 480, 10.2307/1968173 Petkovšek, 1992, Hypergeometric solutions of linear recurrences with polynomial coefficients, J. Symb. Comput., 14, 243, 10.1016/0747-7171(92)90038-6 Petkovšek Petkovšek Petkovšek, 2013, Solving linear recurrence equations with polynomial coefficients, 259 Petkovšek, 1996 Risch, 1969, The problem of integration in finite terms, Trans. Am. Math. Soc., 139, 167, 10.1090/S0002-9947-1969-0237477-8 Schneider, 2001 Schneider, 2004, A collection of denominator bounds to solve parameterized linear difference equations in ΠΣ-extensions, An. Univ. Vest. Timiş., Ser. Mat.-Inform., 42, 163 Schneider, 2005, A new Sigma approach to multi-summation, Adv. Appl. Math., 34, 740, 10.1016/j.aam.2004.07.009 Schneider, 2005, Degree bounds to find polynomial solutions of parameterized linear difference equations in ΠΣ-fields, Appl. Algebra Eng. Commun. Comput., 16, 1, 10.1007/s00200-004-0167-3 Schneider, 2005, Product representations in ΠΣ-fields, Ann. Comb., 9, 75, 10.1007/s00026-005-0242-2 Schneider, 2005, Solving parameterized linear difference equations in terms of indefinite nested sums and products, J. Differ. Equ. Appl., 11, 799, 10.1080/10236190500138262 Schneider, 2007, Symbolic summation assists combinatorics, Sémin. Lothar. Comb., 56, 1 Schneider, 2008, A refined difference field theory for symbolic summation, J. Symb. Comput., 43, 611, 10.1016/j.jsc.2008.01.001 Schneider, 2013, Simplifying multiple sums in difference fields, 325 Schneider, 2015, Fast algorithms for refined parameterized telescoping in difference fields, vol. 8942, 157 Schneider, 2016, A difference ring theory for symbolic summation, J. Symb. Comput., 72, 82, 10.1016/j.jsc.2015.02.002 Schneider, 2017, Summation theory II: characterizations of RΠΣ-extensions and algorithmic aspects, J. Symb. Comput., 80, 616, 10.1016/j.jsc.2016.07.028 Schneider, 2020, Minimal representations and algebraic relations for single nested products, Program. Comput. Softw., 46, 133, 10.1134/S0361768820020103 Singer, 1991, Liouvillian solutions of linear differential equations with Liouvillian coefficients, J. Symb. Comput., 11, 251, 10.1016/S0747-7171(08)80048-X Singer, 1985, An extension of Liouville's theorem on integration in finite terms, SIAM J. Comput., 14, 966, 10.1137/0214069 van der Put, 1997, Galois Theory of Difference Equations, vol. 1666 van Hoeij, 1998, Rational solutions of linear difference equations, 120 van Hoeij, 1999, Finite singularities and hypergeometric solutions of linear recurrence equations, J. Pure Appl. Algebra, 139, 109, 10.1016/S0022-4049(99)00008-0 Zeilberger, 1990, A holonomic systems approach to special functions identities, J. Comput. Appl. Math., 32, 321, 10.1016/0377-0427(90)90042-X Zeilberger, 1991, The method of creative telescoping, J. Symb. Comput., 11, 195, 10.1016/S0747-7171(08)80044-2