[seqfan] Re: help with sequences of form a(n) = Product_{i=1..n} j^i - k^i

[Bob Selcoe]

Hello Seqfans,

Let S be sequences of form a(n) = Product_{i=1..n} j^i - k^i; j>k>=1, n>=1. Generally, given p=(k/j)^n that an outcome will occur at the n-th stage of an infinite process, then r = 1 - a(n)/j^((n^2+n)/2) is the probability that the outcome has occurred up to and including the n-th iteration.  

There are at least 12 OEIS sequences of form S where a(n) = Product_{i=1..n} j^i-1 (i.e., k=1), all with xrefs indicating that j = “q”: A005329 (q=2), A027871 (q=3), A027637 (q=4), A027872 (q=5), A027873 (q=6),A027875(q=7), A027876 (q=8), A027877 (q=9), A027878 (q=10), A027879 (q=11), A027880(q=12).

Each of these entries has a formula given by G.C. Greubel: a(n) = q^(binomial(n+1,2))*(1/q;1/q)_{n}, where (a;q)_{n} is the q-Pochhammer symbol. 

I don't know what “q-Pochhammer symbol” means and there is no further explanation given in these entries. I have found explanations on Wolfram and Wikipedia but do not understand them.

I have submitted A263394 (j=3, k=2), A269576 (j=4, k=3) and A269661 (j=5, k=4) as examples of S where k>1.

Two questions to help resolve some outstanding issues toward publishing these sequences:

1)    It appears that when k/j >= ln(2), the limit ratio of r=1.  Is there a proof?

2)    Empirically it seems possible that q=j/k for S.  Is this accurate - for example, does q = 3/2 for A263394, 4/3 for A269576 and 5/4 for A269661, where the meaning of "q" applies to these other sequences?   

Please feel free to add any observations/explanations to these entries.

Thanks in advance,

Bob Selcoe