ELLIPTIC BERNOULLI q-SERIES AND BERNOULLI NUMBERS


ELLIPTIC BERNOULLI q-SERIES AND BERNOULLI NUMBERS


The well-known Bernoulli numbers B(n), n = 0, 1, 2, 3, 4.... are defined as follows:

u / {e(u) -1} = B(0) + B(1).u/1! + B(2).u^2/2! + B(3). u^3/3! + B(4).u^4/4! + B(5).u^5/5! + B(6).u^6/6! + .... . 
 It is known that B(0) = 1, B(1) = -1/2, B(2) = 1/6, B(3) = 0, B(4) = -1/30, B(5) = 0, B(6) = 1/42.

 We recall from an earlier blogpost the elliptic exponential function E(iu, τ)  := cn (u,τ) + i sn(u,τ). It is clear that lim E(iu, τ) = e(iu) = cos u + i sin u as q approaches 0, where q:=e(iπτ)  . Replacing u by -iu, we get the functions E(u, τ) and e(u), where lim E(u, τ) = e(u) as q approaches 0. We refer to each of  E(iu, τ) and E(u, τ) as the elliptic exponential function, since the context indicates which one is meant.  

 We have  lim  u / {E(u, τ) - 1} = u / {e(u)-1} as q approaches 0.

 Therefore we have 

 u / {E(u, τ) - 1} = β(0) + β(1)u/1! + β(2)u^2/2! + β(3)u^3/3! + β(4)u^4/4! + .... ....

where lim β(n) = B(n) as q approaches 0. 

 The β (n)'s are q-series. We call them elliptic Bernoulli q-series because their constant terms are the Bernoulli numbers B(n). As q approaches 0, all the terms containing positive powers of q collapse to 0 in the limit and we are left with the constant terms which are nothing but the Bernoulli numbers.

  I have found by some rather formidable calculations the β (n)'s , for n=0, 1, 2, 3, 4, 5, 6. They are as follows.

β(0) = 1 

β(1) =  -1/2 

β(2) =  1/6.(1 - 32q + 256q^2 - 1408q^3 + 6144q^4 - 22976q^5 + 76800q^6 - 235264q^7 + 671744q^8 -....),

β(3) = 0

 Continued in the next post.

 Somjit Datta, PhD

 July 8, 2018
Calcutta, India

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