ELLIPTIC EULER q-SERIES AND EULER NUMBERS

As q approaches 0, the following q-series reduce, in the limit, to the well-known Euler
numbers  E(0) = 1, E(1) = 0, E(2) = -1, E(3) = 0, E(4) = 5, E(5) = 0, E(6) = -61, E(8) = 1385, etc.
( i.e the constant term of ε (n) is the Euler number E(n).)

ε (0) = 1,

ε (1) = 0,


ε (2) = -1 - 2 q^9 + 16 q^10 - 76 q^11 + 288 q^12 - 950 q^13 + 2832 q^14 - 7812 q^15

+ 20256 q^16 - .... ....

ε (3) = 0,

ε (4) = 5 - 64 q + 512 q^2 - 2816 q^3 + 12288 q^4 - 45952 q^5 + 153600 q^6 - 470528 q^7 +

1343488 q^8 - 3619114 q^9 + 9280352 q^10 - 22808204 q^11 + 54016448 q^12 - 123808734 q^13

+ 275619936 q^14 - 597712996 q^15 + 1265867712 q^16 - .... ....

ε (5)= 0,

ε (6) = - 61 +1216 q - 13824 q^2 + 119040 q^3 - 856064 q^4 + 5329536 q^5 - 29313024 q^6

+ 144861696 q^7 - 652836864 q^8 + 2716941850 q^9 - 10551777680 q^10 + 38573871772 q^11

- 133685438240 q^12 + 441849019198 q^13 - 1399644240272 q^14 + 4267104538996 q^15

- 12565129215776 q^16 + .... ....

ε (7) = 0,

ε (8) = 1385 - 38784 q + 592896 q^2 - 6490624 q^3 + 56696832 q^4 - 420276480 q^5

+2753171456 q^6 - 16381436928 q^7 +90152042496 q^8 - 464348336722 q^9

+ 2256041335744 q^10 - 10395340234780 q^11 + 45613716208000 q^12 - 191241733994422 q^13

+ 768409826535872 q^14 - 2966950950149076 q^15 + 11036651044668800 q^16 - .... ....