# Hermite Reduction and Creative Telescoping for Hyperexponential Functions

Supplementary material to the paper by Alin Bostan, Shaoshi Chen, Frédéric Chyzak, Ziming Li,and Guoce Xin.

## Maple Implementation:

Please download the following files:

• Maple code: HermiteCT.mm
• For using this package, please see the illustration examples:

## Timings:

We compared the algorithm HermiteTelescoping in our paper to the Maple built-in implementation of the telescoping algorithm by Almkvist and Zeilberger.

The hyperexponential functions we tested are of the form

where m is an positive integer and p, q, a, b, u and v are all irreducible polynomials in x and y over intergers. For simplicity, we choose

The table below shows the timings (in seconds) we got in which:
• ZT: The Maple built-in function Zeilberger in package DEtools.
• HT: The algorithm HermiteTelescoping with computing telescopers only.
• HTC: The algorithm HermiteTelescoping with computing both telescopers and certificates.
• order: The order of computed minimal telescoper.
• Telescoper: The computed minimal telescoper.
• OOM: Maple runs out of memory.
Timings were taken on a Mac OS X machine with 4Gb RAM and 3.06 GHz Core 2 Duo processor.

 Example ( λ, μ, ν, m ) ZT HT HTC order Telescoper example1 (2, 0, 2, 1) 2.16 2.01 3.80 5 tele1 example2 ( 2, 0, 2, 2 ) 2.06 1.98 2.59 5 tele2 example3 ( 3, 0, 2, 1 ) 8.68 6.54 14.01 6 tele3 example4 ( 3, 0, 2, 2 ) 9.23 6.06 13.72 6 tele4 example5 ( 6, 0, 1, 1 ) 44.04 24.39 70.49 7 tele5 example6 ( 6, 0, 1, 2 ) 41.85 22.74 59.50 7 tele6 example7 ( 2, 1, 2, 1 ) 80.69 17.97 47.51 7 tele7 example8 ( 2, 1, 2, 2 ) 66.47 17.25 46.94 7 tele8 example9 ( 2, 1, 2, 1 ) 63.36 17.45 47.94 7 tele9 example10 ( 2, 1, 2, 2 ) 69.69 17.81 47.11 7 tele10 example11 ( 2, 2, 2, 1 ) 1399.2 155.54 570.40 9 tele11 example12 ( 2, 2, 2, 2 ) 1397.7 142.34 510.11 9 tele12 example13 ( 3, 0, 3, 1 ) 151.84 44.07 120.44 8 tele13 example14 ( 3, 0, 3, 2 ) 150.14 43.46 122.36 8 tele14 example15 ( 3, 3, 0, 1 ) 206.90 46.15 165.67 8 tele15 example16 ( 3, 3, 0, 2 ) 207.81 44.95 161.25 8 tele16 example17 ( 3, 2, 1, 1 ) 300.93 60.33 184.71 8 tele17 example18 ( 3, 2, 1, 2 ) 333.75 55.86 176.78 8 tele18 example19 ( 3, 1, 3, 1 ) OOM 361.79 1556.1 10 tele19 example20 ( 3, 1, 3, 2 ) OOM 370.18 1535.7 10 tele20