Skip to main content
MathOverflow
Ira Gessel's user avatar
Ira Gessel's user avatar
Ira Gessel's user avatar
Ira Gessel
  • Member for 14 years, 1 month
  • Last seen this week
  • Brandeis University, Waltham, MA, United States
Profile Activity

822 Actions

All Accepts Posts Badges Comments Revisions Reviews Suggestions
awarded
 co.combinatorics
answered
A hypergeometric identity related to Bessel functions
comment
Padé Approximants of Power Series with Natural Boundaries
The easiest way is to copy and paste. Search for "Gabor Szego" and the first hit is the Wikipedia page with the correct diacritics. (Or just copy from my comment.)
comment
Padé Approximants of Power Series with Natural Boundaries
A minor point, but it's a long umlaut on the o: ő, not ö.
comment
Padé Approximants of Power Series with Natural Boundaries
It's Gábor Szegő.
comment
Is the field of q-series 'dead'?
@kartop_man See, for example, mathworld.wolfram.com/q-Series.html.
comment
Combinatorial meaning of the functional equation for logarithm
See also math.stackexchange.com/questions/3021631/….
comment
One generating function, two-fold sums
Another useful thing to know is that $x+2+2\sqrt{x+1} = (1+\sqrt{x+1})^2$.
awarded
 Necromancer
answered
Fibonacci identity using generating function
revised
Is this a new Fibonacci Identity?
edited body
comment
General term formula for sequences
Expand $1/(1-K(x))= \sum_n K(x)^n$ by the multinomial theorem. See my answer to mathoverflow.net/questions/53384/….
comment
General term formula for sequences
Let $A(x) = \sum_{n=0}^\infty a_n x^n/n!$, where $a_0=1$, and let $K(x) = \sum_{n=1}^\infty k_n x^n/n!$. Then $A(x) = 1/(1-K(x))$.
answered
p-adic valuation of coefficients of generating function
comment
p-adic valuation of coefficients of generating function
By Lagrange inversion, $$g(t) = \sum_{j=0}^\infty \frac{1}{p^j(pj+1)}\binom{pj+1}{j} t^{(p-1)j+1}.$$
comment
How many permutations are there at a given Cayley distance from the identity?
It's worth noting that a permutation in $S_n$ with $k$ cycles has Cayley distance $n-k$. This is why Stirling numbers of the first kind appear.
answered
One generating function, two-fold sums
comment
One generating function, two-fold sums
This doesn't answer your questions, but (2) = (3) is a special case of Pfaff's transformation for the hypergeometric series. More precisely, Pfaff's transformation takes the reversel of (2) to (3).
revised
Eulerian number identity
edited body
comment
Eulerian number identity
Yes, I should have started all my summations at $n=1$, not $n=0$. (I just fixed this.) The reason for taking out the constant term is so that I can write the inner sum as $\sum_{m=0}^{n-1}$, as in the statement of the problem, which doesn't work for $n=0$.
1
…
19 20
21
22 23
…
42