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A holonomic systems approach to special functions identities. (English) Zbl 0738.33001

It is possible to treat the subject of identities satisfied by his hypergeometric functions from many points of view. Lie groups and algebras are very useful for certain problems, addition theorems being one but far from the only one. The author has used other algebraic methods to develop methods to prove and often to derive identities, such as recurrence relations satisfied by polynomial hypergeometric series, or generating functions for them. The present paper uses J. Bernstein’s work on holonomic systems to set up a powerful machine, which is often effective in theory, and sometimes in practice.
Later, he developed other methods which are not as complete in theory, but are much more practical, so that identities of a certain important type can now be treated by one method. Some of the later work was joint with H. Wilf. Much of this works for basic hypergeometric series, as well as hypergeometric series.
Reviewer: R.Askey (Madison)

MSC:

33C20 Generalized hypergeometric series, \({}_pF_q\)

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[1] G. Almkvist, Zeilbergers bevismaskin, Elementa, to appear (in Swedish).; G. Almkvist, Zeilbergers bevismaskin, Elementa, to appear (in Swedish).
[2] G. Almkvist and D. Zeilberger, The method of differentiation under the integral sign, J. Symbolic Comput., to appear.; G. Almkvist and D. Zeilberger, The method of differentiation under the integral sign, J. Symbolic Comput., to appear. · Zbl 0717.33004
[3] Andrews, G. E., Connection coefficient problems and partitions, (Ray-Chaudhuri, D., AMS Proc. Symposia in Pure Mathematics, 34 (1979), Amer. Mathematical Soc: Amer. Mathematical Soc Providence, RI), 1-24 · Zbl 0186.30203
[4] Andrews, G. E., q-Series: Their Development And Applications in Analysis, Number Theory, Combinatorics, Physics, and Computer Algebra, 66 (1986), Amer. Mathematical Soc: Amer. Mathematical Soc Providence, RI, CBMS Ser. · Zbl 0594.33001
[5] Askey, R. A., Math. Rev., 83f:33001 (1983), Review of [60]
[6] Askey, R. A., Preface, (Askey, R. A.; Koornwinder, T. H.; Schempp, W., Special Functions: Group Theoretical Aspects and Applications (1984), Reidel: Reidel Dordrecht) · Zbl 1354.00089
[7] Askey, R. A.; Gasper, G., Inequalities for polynomials, (Baernstein, A., The Bieberbach Conjecture (1986), Amer. Mathematical Soc: Amer. Mathematical Soc Providence, RI) · Zbl 0212.40904
[8] Askey, R. A.; Wilson, J. A., Some Basic Hypergeometric Orthogonal Polynomials that Generalize Jacobi Polynomials, 318 (1985), Amer. Mathematical Soc: Amer. Mathematical Soc Providence, RI, Mem. Amer. Math. Soc. · Zbl 0572.33012
[9] Bailey, W. N., Generalized Hypergeometric Series, 32 (1935), Cambridge Univ. Press: Cambridge Univ. Press London, (reprinted: Hafner, New York, 1964) · Zbl 0011.02303
[10] Bernstein, I. N., Modules over a ring of differential operators, study of the fundamental solutions of equations with constant coefficients, Functional Anal. Appl., 5, 2, 89-101 (1971), (English translation) · Zbl 0233.47031
[11] Bernstein, I. N., The analytic continuation of generalized functions with respect to a parameter, Functional Anal. Appl., 6, 4, 273-285 (1972), (English translation) · Zbl 0282.46038
[12] Björk, J.-E., Rings of Differential Operators (1979), North-Holland: North-Holland Amsterdam
[13] Björk, J.-E., Differential systems on algebraic manifolds (1984), Univ. Stockholm, preprint
[14] Buchberger, B., Grobner bases—an algorithmic method in polynomial ideal theory, (Bose, N. K., Multidimensional System Theory (1985), Reidel: Reidel Dordrecht), Chapter 6 · Zbl 0587.13009
[15] Cipra, B., How the grinch stole mathematics, Science, 245, 595 (1989)
[16] de Brange, L., A proof of the Bieberbach conjecture, Acta Math., 154, 137-152 (1985) · Zbl 0573.30014
[17] Ehrenpreis, L., Fourier Analysis in Several Variables (1970), Wiley: Wiley New York · Zbl 0195.10401
[18] L. Ehrenpreis, Hypergeometric functions, preprint, Temple Univ., PA.; L. Ehrenpreis, Hypergeometric functions, preprint, Temple Univ., PA. · Zbl 0686.33010
[19] Fasenmayer, M. C., Some generalized hypergeometric polynomials, Bull. Amer. Math. Soc., 53, 806-812 (1947) · Zbl 0032.15402
[20] Foata, D., Combinatoire des identités sur les polynômes orthogonaux, (Proc. Internat. Congress of Math. (1983), Varsovie), 1541-1553, Warsaw, 1983
[21] Galligo, A., Some algorithmic questions on ideals of differential operators, (Cavines, B. F., EUROCAL ’85, Vol. 2, 204 (1985), Springer: Springer Berlin), Lecture Notes in Comput. Sci. · Zbl 0634.16001
[22] Gasper, G., Summation, transformation, and expansion formulas for bibasic series, Trans Amer. Math. Soc., 312, 257-277 (1989) · Zbl 0664.33010
[23] G. Gasper and M. Rahman, An indefinite bibasic summation formula and some quadratic, cubic, and quartic summation and transformation formulas, Canad. J. Math., to appear.; G. Gasper and M. Rahman, An indefinite bibasic summation formula and some quadratic, cubic, and quartic summation and transformation formulas, Canad. J. Math., to appear. · Zbl 0707.33009
[24] Gasper, G.; Rahman, M., Basic Hypergeometric Series (1990), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0695.33001
[25] Gel’fand, I. M., General theory of hypergeometric functions, Soviet Math. Dokl., 33, 3, 573-577 (1986), translated in · Zbl 0645.33010
[26] Gel’fand, I. M.; Gel’fand, S. I., Generalized hypergeometric equations, Soviet Math. Dokl., 33, 3, 643-646 (1986), translated in · Zbl 0634.58030
[27] Gel’fand, I. M.; Graev, M. I., A duality theorem for general hypergeometric functions, Soviet Math. Dokl., 34, 1, 9-13 (1987), translated in · Zbl 0619.33006
[28] Gel’fand, I. M.; Zelevinskii, A. V., Algebraic and combinatorial aspects of the general theory of hypergeometric functions, Functional Anal. Appl., 20, 3, 183-197 (1986), (English translation) · Zbl 0573.22008
[29] Gessel, I., Two theorems on rational power series, Utilitas Math., 19, 247-254 (1981) · Zbl 0477.13008
[30] Gessel, I., Counting Latin rectangles, Bull. Amer. Math. Soc. (N.S.), 16, 79-82 (1987) · Zbl 0617.05015
[31] I. Gessel, Symmetric functions and P-recursiveness, preprint, Brandeis Univ.; I. Gessel, Symmetric functions and P-recursiveness, preprint, Brandeis Univ. · Zbl 0704.05001
[32] Gessel, I.; Stanton, D., Strange evaluations of hypergeometric series, SIAM J. Math. Anal., 13, 295-308 (1982) · Zbl 0486.33003
[33] B. Gosper, Letters to R. Askey and D. Stanton, private communication.; B. Gosper, Letters to R. Askey and D. Stanton, private communication.
[34] I.P. Goulden, Exact values for degree sums over strips of Young diagrams, preprint, Univ. Waterloo.; I.P. Goulden, Exact values for degree sums over strips of Young diagrams, preprint, Univ. Waterloo. · Zbl 0723.05118
[35] D. Gouyou-Beauchamps, Standard Young tableaux of height 4 and 5, preprint.; D. Gouyou-Beauchamps, Standard Young tableaux of height 4 and 5, preprint.
[36] Knuth, D. E., (Fundamental Algorithms, The Art of Computer Programming, 1 (1973), Addison-Wesley: Addison-Wesley Reading, MA)
[37] Labelle, J., Tableau d’Askey, (Brezinski, C., Polynômes Orthogonaux et Applications, 1171 (1985), Springer: Springer Berlin), vii, Bar-le-Duc, 1984, Lecture Notes in Math.
[38] Lipshitz, L., The diagonal of a D-finite power series is D-finite, J. Algebra, 113, 373-378 (1988) · Zbl 0657.13024
[39] L. Lipshitz, D-finite power series, J. Algebra, to appear.; L. Lipshitz, D-finite power series, J. Algebra, to appear. · Zbl 0695.12018
[40] Macdonald, I. G., Some conjectures for root systems, SIAM J. Math. Anal., 13, 91-143 (1982) · Zbl 0498.17006
[41] Miller, W., Symmetry and Separation of Variables, 4 (1977), Addison-Wesley: Addison-Wesley London, Encyclopedia Math. Appl.
[42] Milne, S. C., Hypergeometric series well poised in SU(n) and a generalization of Biedenharn’s G function, Adv. in Math., 36, 169-211 (1980) · Zbl 0451.33010
[43] Palamadov, V. P., Linear Differential Operators with Constant Coefficients, 168 (1970), Springer: Springer Berlin, Grundlehren Math. Wiss.
[44] Perlstadt, M., Recurrences for sums of powers of binomial coefficients, J. Number Theory, 27, 304-309 (1987) · Zbl 0626.10010
[45] Rainville, E. D., Special Functions (1960), Macmillan: Macmillan New York, (reprinted: Chelsea, New York, 1971) · Zbl 0050.07401
[46] Regev, A., Asymptotic values for degrees associated with strips of Young diagrams, Adv. in Math., 41, 115-136 (1981) · Zbl 0509.20009
[47] Riordan, J., An Introduction to Combinatorial Analysis (1980), Princeton Univ. Press: Princeton Univ. Press Princeton, NJ, (originally published by (Wiley, New York, 1958)) · Zbl 0436.05001
[48] Roy, R., Binomial identities and hypergeometric series, Amer. Math. Monthly, 94, 37-46 (1987) · Zbl 0621.33005
[49] Stafford, J. T., Module structure of Weyl algebras, J. London Math. Soc. (2), 18, 429-442 (1978) · Zbl 0394.16001
[50] Stanley, R., Differentiably finite power series, European J. Combin., 1, 175-188 (1980) · Zbl 0445.05012
[51] Sylvester, J. J., A method of determining by mere inspection the derivatives from two equations of any degree, Collected Works I, 16, 54-57 (1840)
[52] N. Takayama, Holonomic solutions of Weisner’s operator, preprint, Tokushima Univ.; N. Takayama, Holonomic solutions of Weisner’s operator, preprint, Tokushima Univ. · Zbl 0689.58003
[53] Takayama, N., Grobner basis and the problem of contiguous relations, Japan J. Appl. Math., 6, 147-160 (1989) · Zbl 0691.68032
[54] Tukey, J. W., Amer. Statist., 40, 74 (1986)
[55] van der Poorten, A., A proof that Euler missed… Apery’s proof of the irrationality of ζ(3), Math. Intelligencer, 1, 195-203 (1979) · Zbl 0409.10028
[56] van der Waerden, B. L., Modern Algebra, I and II (1940), Frederick Ungar: Frederick Ungar New York · Zbl 0022.29801
[57] Wilf, H. S., What is an answer?, Amer. Math. Monthly, 89, 289-292 (1982) · Zbl 0485.05001
[58] H.S. Wilf and D. Zeilberger, Towards computerized proofs of identities, Bull. Amer. Math. Soc., to appear.; H.S. Wilf and D. Zeilberger, Towards computerized proofs of identities, Bull. Amer. Math. Soc., to appear. · Zbl 0718.05010
[59] H.S. Wilf and D. Zeilberger, Rational functions certify combinatorial identities, J. Amer. Math. Soc., to appear.; H.S. Wilf and D. Zeilberger, Rational functions certify combinatorial identities, J. Amer. Math. Soc., to appear. · Zbl 0695.05004
[60] Zeilberger, D., Sister Celine’s technique and its generalizations, J. Math. Anal. Appl., 85, 114-145 (1982) · Zbl 0485.05003
[61] D. Zeilberger, A fast algorithm for proving terminating hypergeometric identities, preprint.; D. Zeilberger, A fast algorithm for proving terminating hypergeometric identities, preprint. · Zbl 0701.05001
[62] D. Zeilberger, The method of creative telescoping, preprint.; D. Zeilberger, The method of creative telescoping, preprint. · Zbl 0738.33002
[63] D. Zeilberger, The method of creative telescoping for q-series, in preparation.; D. Zeilberger, The method of creative telescoping for q-series, in preparation. · Zbl 0738.33002
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