Begin with the program RaduRK.

<< RaduRK`

math4ti2: Mathematica interface to 4ti2 (http://www.4ti2.de/)
Copyright (C) 2017, Ralf Hemmecke <ralf@hemmecke.org>
Copyright (C) 2017, Silviu Radu <sradu@risc.jku.at>

RaduRK: Ramanujan--Kolberg Program
Copyright (C) 2019, Nicolas Allen Smoot <nsmoot@risc.uni-linz.ac.at>
Research Institute for Symbolic Computation
Johannes Kepler Universität, Linz

Next, establish the two key variables q and t.

SetVar1[q]

SetVar2[t]

For each case we work over a modular curve (Subscript[X, 0])(N), with N a whole number.

Take an arithmetic function defined by a generating function which can be constructed with the integer M and the integer-valued vector r.

Finally, take the linear progression mn+j, as n goes over the nonnegative integers.

Start with the ordinary partition function, with the progression 5n+j, j=0,1,2,3,4.  We work over N=5.

RKDelta[5,1,{-1},5,4]

RKDelta[5,1,{-1},5,1]

RKDelta[5,1,{-1},5,3]

What about 25n+24?

RKDelta[5,1,{-1},25,24]

We examine next the ordinary partition function, with the progression 7n+j, j=0,1,2,3,4,5,6.  We work over N=7.

RKDelta[7,1,{-1},7,5]

RKDelta[7,1,{-1},7,1]

RKDelta[7,1,{-1},7,2]

What about 49n+47?

RKDelta[7,1,{-1},49,47]

We examine next Kolberg’s Identity for p(3n+j), N=9.

RKDelta[9,1,{-1},3,1]

We examine next the ordinary partition function, with the progression 11n+6.  Start with N=11.

RKDelta[11,1,{-1},11,6]

Try N=22.

Timing[RKDelta[22,1,{-1},11,6]]

We examine next a function related to broken 
2-diamond partitions, with the progression 25n+24.  We take N=10.

RKDelta[10,10,{2,1,0,-1},25,14]

RKDelta[10,10,{-3,1,1,-1},25,14]

We examine a function related to Rogers--Ramanujan subpartitions, with the progression 25n+24.  We work over N=20.  Notice the witness identity for divisibility by 5, as discovered by Choi, Kim, and Lovejoy.

Timing[RKDelta[20,4,{-3,5,-2},25,24]]

We examine generalized 2-colored Frobenius subpartitions, with the progression 5n+3.  We work over N=20.  Notice the witness identity for divisibility by 5 (proved by Andrews), in accordance with the Andrews--Sellers conjecture (proved by Paule and Radu).

Timing[RKDelta[20,4,{-4,5,-2},5,3]]

To test our additional procedures, we examine some identities related to 5-Regular Bipartitions.  First, we use the standard RKDelta.

Timing[RKDelta[20,5,{-2,2},4,3]]

Now we will construct the algebra basis for eta quotients (with a pole only at infinity) over Gamma_0(N).  From here, we use RKDeltaMan.

G20=etaGenerators[20];

A20=AB[G20]

Timing[RKDeltaMan[20,5,{-2,2},4,3,A20]]

Next, we incorporate a computational bound for k0 (see Theorem 5 of our implementation paper) by using RKDeltaE.

Timing[RKDeltaE[20,5,{-2,2},4,3]]

Finally, we use RKDeltaEMan, with the separately computed algebra basis, with k0 accounted for.

Timing[RKDeltaEMan[20,5,{-2,2},4,3,A20]]