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**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.

Reviewer: R.Askey

### MSC:

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

### Keywords:

evaluation of integrals
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\textit{G. Almkvist} and \textit{D. Zeilberger}, J. Symb. Comput. 10, No. 6, 571--591 (1990; Zbl 0717.33004)

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