A User's Manual for the Maple program Theta3Romik.txt implementing Dan Romik's article "The Taylor coefficients of the Jacobi θ3 constant. "

By Doron Zeilberger

Written: July 16, 2018

This is a user's manual to the Maple package Theta3Romik.txt, that implements Dan Romik's elegant article "The Taylor coefficients of the Jacobi θ3 constant

# Instructions for installing

Save the package as Theta3Romik.txt . Stay in the same directory, go into Maple (typing maple, or xmaple), and then type

To get a list of the main procedures type :

ezra();

To get help with a specific procedure, type:

ezra(ProcedureName);

# Sample Input and Output

• If you want to study Conjecture 13

the input file generates the output file.

• For an extension of the Appendix listing the first 200 terms of the Romik sequence d(n) the input file generates the output file.

# Short Descriptions of the Main procedures

• cAB(A,B,t,K): inputs polynomials A and B of the variable t (with no constant terms), and a positive integer K finds the polynomial in t, C, of degree K, such that C(A(t))-B(t) has low-degree>=K+1, in other words, the Taylor expansion of C(A(t))-B(t) starts at t^(K+1). For example, try:
cAB(t+t^2,t+t^3,t,10);

• dSeq(K): the first K terms of the Dan Romik sequence d(n), done directly. For example, to get the first 20 terms, type:

dSeq(20);

• dSeqF(K): the first K terms of the Dan Romik sequence d(n), done done via (the much slower (for large K)) Theorem 7 . For example, type:

dSeqF(20);

• dSeqPC(K): the Pre-Computed, first K terms of the Dan Romik sequence d(n), for K ≤ 200. Of course it is much faster . For example, to immediately get the first 200 terms of the sequence, type

dSeqPC(200);

• PerStory(K): the conjectured periods (for primes 1 mod 4) or starting to be 0 (for primes 3 mod 4) using K values of the Romik sequence.

The output to PerStory(200) is given in the above-mentioned output file.

• rnkSeq(K):the list of lists, let's call it L, such that L[n][k] is r(n,k), 1<=k<=n. Try:

rnkSeq(30);

Maple programs of Doron Zeilberger