The C-finite Ansatz Meets the Holonomic Ansatz

The C-finite Ansatz Meets the Holonomic Ansatz

By
Shalosh B. Ekhad and Doron Zeilberger
.pdf .ps .tex

Posted: Dec. 21, 2015.
[Exclusively published in the Personal Journal of Shalosh B. Ekhad and Doron Zeilberger]

Could you imagine a paper published nowadays (or even two thousands years ago), entitled
"A new proof of the identity 134 Times 431 Equals 57754" ?
or, nowadays (or even two hundred and fifty years ago) entitled
"A New and Elegant proof of the algebraic identity a³-b³=(a-b)(a²+2ab+b²)" ?
Yet many articles, regarding identities among special functions, for which there exist known fully-implemented algorithms that can handle them automatically (often very fast), are still published today, often assisted by computer-algebra software, but the authors and editors do not realize that they can be fully generated, without any human "help" (except for entering the data).

Maple Package

CfiniteIntegral.txt, Finds recurrences for sequences defined by integrals of powers of sequences of families of polynomials in x (like the Chebyshev) that satisfy a linear recurrence equation with coefficients that are polynomials in x (but do not depend on the index n)

Some Input and Output files for the Maple package CfiniteIntegral.txt

If you want to fully automaticaly rederive the results in Seon-Hong Kim's article, "On some integrals involving Chebyshev polynomials", Ramanujan J. 38 (2015), 629-639
the input yields the output

If you want to see a verbose version of the above
the input yields the output

If you want to see some random examples of procedure IntC
the input yields the output

If you want to see a verbose version of the above
the input yields the output

If you want to see recurrences for the sequences given by integrals from 0 to 1 of the powers of Chebyshev polynomials of the First kind, from the first through the fourth
the input yields the output

If you want to see a verbose version of the above
the input yields the output

If you want to see recurrences for the sequences given by integrals from 0 to infinity of the powers of Chebyshev polynomials of the First kind, times exp(-x), from the first through the third
the input yields the output

If you want to see a verbose version of the above
the input yields the output

If you want to see inhomogeneous differential equations for the ordinary generating functions of the sequences given by integrals from -1 to 1 of the powers of the Chebyshev polynomials of the First kind, times 1/(1+x²), from the first through the fourth
the input yields the output

If you want to see a verbose version of the above
the input yields the output

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