## The method of differentiating under the integral sign.(English)Zbl 0717.33004

The second author and his computer are trying to automate the problem of summing series, and in this paper he is joined by the first author and his computer in trying to do the same for evaluating definite integrals. For single series of hypergeometric type the progress has been striking. The results for integrals are less so, partly I think because the right class of integrals has not yet been found. While series whose term ratio is rational is a reasonable definition for hypergeometric series, having an integrand whose logarithmic derivative is rational is too narrow. For example, the Barnes beta integral does not fit in this class. In addition to differentiating under the integral sign, as in the title, the authors do some differencing under the integral sign, as in the title, the authors do some differencing under the integral sign. This is a very promising area, but it is likely the difference parameter should be real rather than discrete and a particular solution needs to be picked out. The general setting for differentiating under the integral is J. Bernstein’s class of holonomic functions.

### MSC:

 33C60 Hypergeometric integrals and functions defined by them ($$E$$, $$G$$, $$H$$ and $$I$$ functions)

### Keywords:

evaluation of integrals

DEtools ; Maple
Full Text:

### References:

 [1] Almkvist, G., Zeilberger, D. (to appear). A MAPLE program that finds, and proves, recurrences and differential equations satisfied by hyperexponential definite integrals. SIGSAM Bulletin. [2] Bailey, W.N., Generalized hypergeometric series, (), Reprinted · Zbl 0011.02303 [3] Bernstein, I.N., Modules over a ring of differential operators, study of the fundamental solutions of equations with constant coefficients, Functional analysis and its applications, Functional analysis and its applications, 5, 89-101, (1971), (English translation) · Zbl 0233.47031 [4] Björk, J.E., Rings of differential operators, (1979), North Holland Amsterdam [5] Buchberger, B., Gröbner bases—an algorithmic method in polynomial ideal theory, () · Zbl 0587.13009 [6] Coxeter, H.S.M., Twelve geometric essays, (1968), Southern Illinois University Press Carbondale · Zbl 0176.17101 [7] Davenport, J.H.; Siret, Y.; Tournier, E., Computer algebra, (1988), Academic Press London · Zbl 0679.68058 [8] Edwards, J., () [9] Ehlers, F., The Weyl algebra, () [10] Feynman, R.P., (), 71-72, (1985), W.W. Norton, (Published in paperback) [11] Foata, D., A combinatorial proof of the mehler formula, J. comb. theory, ser. A, 24, 250-259, (1978) [12] Foata, D., Some Hermite polynomial identities and their combinatorics, Advances in appl. math., 2, 250-258, (1981) · Zbl 0475.33006 [13] Foata, D.; Garsia, A.M., A combinatorial approach to the mehler formulas for Hermite polynomials, (), 163-179 [14] Galligo, A., Some algorithmic questions on ideals of differential operators, () · Zbl 0634.16001 [15] Gosper, R.W., Decision procedure of indefinite summation, Proc. nat. acad. sci. USA, 75, 40-42, (1978) · Zbl 0384.40001 [16] Graham, R.; Patashnik, O.; Knuth, D.E., Concrete mathematics, (1989), Addison Wesley Reading [17] Herr, J.P., Lehrbuch der hoeren Mathematik, 2, 356, (1857), Wien [18] Hochstadt, H., Special functions of mathematical physics, (1961), Holt, Rinehart and Winston New York · Zbl 0102.05501 [19] Lafon, J.C., Summation in finite terms, () · Zbl 0495.68036 [20] Lipshitz, L., The diagonal of a D-finite power series is D-finite, J. algebra, 113, 373-378, (1988) · Zbl 0657.13024 [21] Meelzak, Z.A., Companion to concrete mathematics, (1973), John Wiley New York [22] Norman, A.C., Integration in finite terms, () · Zbl 0494.68042 [23] Rainville, E.D., Special functions, Special functions, (1971), Bronx Chelsea, (Reprinted by) · Zbl 0231.33001 [24] Risch, R.H., The solution of the problem of integrating in finite terms, Bull. AMS, 76, (1970) · Zbl 0196.06801 [25] Takayama, N., An approach to the zero recognition problem by the buchberger algorithm, (1989), Kobe University, Preprint [26] Trager, B.M., On the integration of algebraic functions, () · Zbl 0606.68032 [27] Wilf, H.S.; Zeilberger, D., Towards computerized proofs of identities, Bulletin of the amer. math. soc., 22, 77-83, (1990) · Zbl 0718.05010 [28] Zeilberger D. (to appear). A holonomic systems approach to special function identities. J. Comp. and Appl. Math. · Zbl 0738.33001 [29] Zeilberger, D., A fast algorithm for proving terminating hypergeometric identities, Discrete math., 80, 207-211, (1990) · Zbl 0701.05001 [30] Zeilberger D. (to appear). The method of creative telescoping. J. Symbolic Comp. · Zbl 0738.33002
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