All Questions
4
questions with no upvoted or accepted answers
2
votes
0
answers
63
views
How to manipulate this summation in the easiest way possible?
$$
D =
\sum_{k=c}^{n}\sum_{j=0}^{k-c}[{k-c \choose j}\ln^{k-c-j}(g(x))[\ln(g) f'(x) f_c^{(j)} X_{n,k(f\rightarrow g)^c} + f_{c}^{(j)} X_{n,k(f \rightarrow g)^{c}}' + \frac{d}{dx}[f_c^{(j)}] X_{n,k(f \...
- 1,957
1
vote
1
answer
66
views
Values for which this sum can be defined in terms of known constants in a closed form
I'm interested in the sum,
$$\sum_{n=1}^\infty\frac{\zeta(2n)\Gamma(2n)}{\Gamma(2k+2n+2)}x^{2n}$$
Otherwise written as
$$\sum_{n=1}^\infty\frac{\zeta(2n)}{(2n)(2n+1)\cdots(2n+2k+1)}x^{2n}$$
I am ...
- 3,557
0
votes
0
answers
34
views
Variants of geometric sum formula
I know there are closed formulas for sums of the form $\sum_{k=0}^n k^sr^k$
and $\sum_{k=0}^n r^{2n}$ or $\sum_{k=0}^n r^{2n+1}$.
(See https://en.wikipedia.org/wiki/Geometric_series#Sum)
From Sum of ...
0
votes
0
answers
95
views
Closed form for $Q_{m}(x)=\sum _{n=0}^{\infty } \frac{ (-1)^n}{n+m}\frac{x^{n}}{n!}$
I've just stumbled upon this
$$
Q_{m}(x)=\sum _{n=0}^{\infty } \frac{ (-1)^n}{n+m}\frac{x^{n}}{n!}
$$
and i'd like to know if it has a closed form.
Note that $m \geq1 $ is an integer.
Thanks.
- 5,627