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

Personal Journal of Shalosh B. Ekhad and Doron Zeilberger