Sum of odd powers 1^m+3^m+5^m+...+(2n+1)^m

[Rfael69]

how can i evaluate the sum $$
1^m+3^m+5^m+ ....................(2n+1)^{m} $$



for the case of the normal sum $$ 1^m +2^m +........................+n^, $$ for positive 'm' i know they are related to the Bernoulli Polynomials

 

[fresh_42]

You want to calculate $\sum_{k=0}^m(2k+1)^m.$ Next, $(2k+1)^m=\sum_{j=0}^m \binom{m}{j}(2k)^j.$ I would start with that.