Supplement (I) to:
“An Implementation of Radu’s Ramanujan--Kolberg Algorithm”
Nicolas Allen Smoot
Research Institute for Symbolic Computation
(RISC), Linz

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) 2020, Nicolas Allen Smoot <nicolas.smoot@risc.jku.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 an integer M and an integer-valued vector r indexed over the divisors of M.

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

Classical Identities

Section 3.1

Start with the ordinary partition function, with the progression 5n+4.  If we work over N=5, we retrieve Ramanujan’s famous identity.

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

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

Section 3.2

We examine the identities discovered by Kolberg and Zuckerman.

To begin with, we compute an identity by Zuckerman.

RK[13,1,{-1},13,6]

Kolberg originally computed the following identities.

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

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

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

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

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

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

Finally, we examine p(2n+j).  The recommended value of N is 16.

minN[1,{-1},2,1]

RK[16,1,{-1},2,1]

However, Kolberg discovered an identity for N=8.  Attempting this substitution, surely enough we get:

RK[8,1,{-1},2,1]

p(11n+6)

Section 3.3.

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

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

We find no identity over (Subscript[X, 0])(11).  This is in part why no simple witness identity for p(11n+6) exists, in contrast to the cases of p(5n+4), p(7n+5).

Let’s next try N=22.

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

We get an identity in which our membership coefficients are integers divisible by 11... although it is not at all obvious that the elements of our algebra basis expand into integer power series.

Broken 2-Diamond Partitions

Section 3.4.

We examine next the broken 
2-diamond partition function, with the progressions 25n+24, 25n+14.

Radu was able to prove some congruences for this function by computing RK identities for a related function.

With the complete implementation, we can compute an RK identity by directly manipulating the broken 
2-diamond partition function itself.  We take N=10.

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

A Restricted Overpartition Function

Section 3.5.3.

We give several results originally due to Munagi and Sellers.  For our identities related to the unrestricted overpartition function, see the second Mathematica supplement.

RK[6,6,{-2,1,1,-1},3,1]

RK[6,6,{-2,1,1,-1},3,2]

RK[6,6,{-2,1,1,-1},27,26]

RK[6,18,{-2,1,0,0,1,-1},27,24]

Identities by Baruah and Sarmah

Section 3.6.

Baruah and Sarmah.

RK[4,1,{8},4,3]

minN[1,{-4},4,3]

RK[8,1,{-4},4,3]

minN[1,{-8},4,3]

minN[1,{-8},2,1]

RK[2,1,{-8},2,1]

RK[2,1,{-8},4,3]

minN[1,{-2},5,2]

RK[5,1,{-2},5,2]

RK[5,1,{-2},5,3]

minN[1,{-4},5,3]

RK[5,1,{-4},5,3]

RK[4,1,{-8},8,7]

5-Regular Bipartitions

Section 3.7.

5-Regular Bipartitions (Wang)

minN[5,{-2,2},4,3]

RK[10,5,{-2,2},4,3]

RK[20,5,{-2,2},4,3]

minN[5,{-2,2},5,2]

RK[5,5,{-2,2},5,2]

RK[5,5,{-2,2},5,3]

minN[5,{-2,2},20,7]

RK[10,5,{-2,2},20,7]

Ramanujan’s τ Function

Section 3.8.

We examine Ramanujan Tau Function (Ramanujan, Banerjee).

RK[7,1,{24},7,6]

minN[1,{24},112,55]

RK[14,1,{24},112,55]

Rogers--Ramanujan Subpartitions

Section 3.9.

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

Testing Additional Procedures

RK

To test our additional procedures, we examine some identities related to 5-Regular Bipartitions.  First, we use the standard RK procedure.  We will time each of these cases.

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

RKMan

Now we will construct the algebra basis for eta quotients (with a pole only at infinity) over (Subscript[X, 0])(N).  From here, we use RKMan.

G20=etaGenerators[20];

A20=AB[G20]

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

RKE

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

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

RKManE

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

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