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Results tagged with sequences-and-series
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user 10744
for questions about sequences and series, e.g. convergence, closed form expressions, etc. Note that there is a different tag for spectral sequences, and also note that MathOverflow is not for homework. Please consider consulting the online encyclopedia for integer sequences, if you are trying to identify a given sequence that you have found in your research.
50
votes
Accepted
New binomial coefficient identity?
In terms of hypergeometric series, the sum is $_3F_2(-n, 1+n, 1/2;1,3/2;1)$ and the identity is a special case of Saalschütz's theorem (also called the Pfaff-Saalschütz theorem), one of the standard h …
- 17k
32
votes
Accepted
Integrality of a sequence formed by sums
Let $A(x) = \sum_{n=1}^\infty a_n x^n$ and let
$$S(x) = \sum_{k=0}^\infty (7k+8)\frac{(3k+1)!}{k!\,(2k+3)!} x^k.$$
Then the formula for $a_n$ gives
$A(x) = R(x)S(x)$,
where
$$R(x) = \frac{1}{3}\biggl( …
- 17k
24
votes
power series of the reciprocal... does a recursive formula exist for the coefficients
Without loss of generality we can take $b_0$ to be 1, since
\begin{equation*}\sum_{n=0}^\infty b_n x^n = b_0\biggl( 1+\sum_{n=1}^\infty (b_n/b_0)x^n\biggr).
\end{equation*}
Then for $b_0=1$ we have
\ …
- 17k
23
votes
Does the formal power series solution to $f(f(x))= \sin( x) $ converge?
This is also a comment. There's another reasonably efficient way to do this sort of computation. Let $L$ be the linear operator on formal power series defined by $L(g) = g(\sin x)$. (Instead of $\sin …
- 17k
22
votes
Fibonacci series captures Euler $e=2.718\dots$
More generally,
$$\sum_{k=0}^\infty F_{n+k} \frac{x^k}{k!} = e^x\sum_{k=0}^\infty F_{n-k}\frac{x^k}{k!},\tag{1}$$
which is equivalent to Will Sawin's identity.
Similarly,
$$e^x\sum_{k=0}^\infty F_{n+k …
- 17k
14
votes
Accepted
Series involving factorials
The sum
$$\sum_{k=0}^\infty \frac{(a+k)!\,(b+k)!}{k!\,(a+b+c+k+1)!}z^k.$$
is not only a generalized hypergeometric series; it's the original ungeneralized Gauss hypergeometric series,
$$\frac{\Gamma(a …
- 17k
10
votes
Accepted
Identity with Pochhammer and harmonic numbers
Here's a sketch of a proof using "creative telescoping."
Let
$$T(i,j) = \frac{j!^2}{(\tfrac12)_j^2}\cdot \frac{(\tfrac12)_i^2}{i!^2}\frac1{j-i}.$$
Since the identity holds for $j=1$, it suffices to …
- 17k
10
votes
Accepted
Sum of products of exponentials and polynomials
Let $S(N,a,k)= \sum_{n=0}^N a^n n^k$. Multiplying by $x^k/k!$ and summing on $k$ gives the exponential generating function
$$\sum_{k=0}^\infty S(N,a,k) \frac{x^k}{k!}=
\frac{(ae^x)^{N+1}-1}{ae^x-1}.$$ …
- 17k
9
votes
Method to evaluate an infinite sum of ratio of Gamma functions (how does Mathematica do it?)
This is a special case of Watson's Theorem
$$\def\h{\frac{1}{2}}
\def\g#1{\Gamma(#1)\,}
{}_3F_2\left({a,\ b,\ c\atop\h a+\h b+\h, 2c }\biggm| 1 \right)
=\frac{\g\h\g{c+\h}\g{\h a+\h b +\h}\g{c+\h -\ …
- 17k
8
votes
Accepted
Infinite sum of Laguerre polynomials: is $\sum_{k=0}^{\infty}\frac{n!}{(k+n)!}x^kL_n^k(x)^2=...
Everything becomes simpler if add some parameters and start the sum at $k=-n$ instead of $k=0$. Note that if $k$ is a negative integer with $-n\le k \le -1$ then $L_n^k(x)$ is a polynomial in $x$ divi …
- 17k
7
votes
Recreation with Catalan
Here's a sketch of a generating function proof. Recall that
\begin{equation*}
\sum_{m=0}^\infty \binom{2m+k}{m} x^m =\frac{c(x)^k}{\sqrt{1-4x}},
\end{equation*}
where $c(x) = \sum_{n=0}^\infty C_n x^n …
- 17k
7
votes
Accepted
$R$-recursion for the A143017
We proceed by "guessing" a generating function for $R(n,q)$ and verifying that it has the right properties.
According to https://oeis.org/A143017, the generating function $G= \sum_{n=1}^\infty a(n) x^ …
- 17k
6
votes
Accepted
Closed form for the A110501 (unsigned Genocchi numbers (of first kind) of even index)
This is a known result. To quote from Richard Stanley's Enumerative Combinatorics, Volume 2, second edition, solution to problem 8(e) of Chapter 5, page 115: This is equivalent to a conjecture of J. M …
- 17k
5
votes
Non-arithmetic proof of the integrality of a rational expression
The integrality of the coefficients of $(1-k^2x)^{-1/k}$ follows from the integrality of the coefficients of the generalized Catalan number generating function
$$c_k(x)=\frac{1-(1-k^2 x)^{1/k}}{kx},$$ …
- 17k
5
votes
5 different ways to define the same family of integer sequences
Here is a proof that $a_1(n, p, q) = a_2(n,p,q)$.
The generating function for the Stirling numbers of the second kind is
$$\sum_{n=i}^\infty {n\brace i}\frac{x^n}{n!}= \frac{(e^x-1)^i}{i!}.$$
Also
$$\ …
- 17k