ELLIPTIC BERNOULLI q-SERIES AND BERNOULLI NUMBERS (contd.)


ELLIPTIC BERNOULLI q-SERIES AND BERNOULLI NUMBERS (contd.)

Continued from the last post.

β(4) = -1/30.(1 + 344q - 5376q^2 + 67200q^3 - 587776q^4 + 4060224 q^5 - 23632896 q^6 + 

 120578304q^7 - 553598976q^8 + ....),

β(5) = 0,
β(6) = 1/42.(1 - 108q + 54912q^2 - 873152q^3 + 10997760q^4 - ....)

 β(7) = 0,

 β(8) = -1/30.(1 + 1288q - 324416q^2 + 8416512q^3 - 142155776q^4 + ....)

    The coefficients are getting progressively larger and larger, rendering further calculation prohibitively difficult.  The calculations for β(6) and β(8)  were so formidable that I could not progress beyond q^4. 
But the general pattern is clear.
   
  As promised, the Bernoulli numbers 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 have been shown to be the constant terms of these q-series. As q approaches 0, β(n) reduces to B(n) in the limit.

Somjit Datta, PhD

July 8, 2018
Calcutta, India

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