Catalan Moments

Mathematics – Number Theory

Scientific paper

Rate now

  [ 0.00 ] – not rated yet Voters 0   Comments 0

Details

22 pages

Scientific paper

This paper is essentially devoted to the study of some interesting relations among the well known operators $I^{(x)}$ (the interpolated Invert), $L^{(x)}$ (the interpolated Binomial) and Revert (that we call $\eta$). We prove that $I^{(x)}$ and $L^{(x)}$ are conjugated in the group $\Upsilon(R)$. Here $R$ is a commutative unitary ring. In the same group we see that $\eta$ transforms $I^{(x)}$ in $L^{(-x)}$ by conjugation. These facts are proved as corollaries of much more general results. Then we carefully analyze the action of these operators on the set $\mc{R}$ of second order linear recurrent sequences. While $I^{(x)}$ and $L^{(x)}$ transform $\mc{R}$ in itself, $\eta$ sends $\mc{R}$ in the set of moment sequences $\mu_n(h,k)$ of particular families of orthogonal polynomials, whose weight functions are explicitly computed. The moments come out to be generalized Motzkin numbers (if $R=\zz$, the Motzkin numbers are $\mu_n(-1,1)$). We give several interesting expressions of $\mu_n(h,k)$ in closed forms, and one recurrence relation. There is a fundamental sequence of moments, that generates all the other ones, $\mu_n(0,k)$. These moments are strongly related with Catalan numbers. This fact allows us to find, in the final part, a new identity on Catalan numbers by using orthogonality relations.

No associations

LandOfFree

Say what you really think

Search LandOfFree.com for scientists and scientific papers. Rate them and share your experience with other people.

Rating

Catalan Moments does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Catalan Moments, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Catalan Moments will most certainly appreciate the feedback.

Rate now

     

Profile ID: LFWR-SCP-O-524308

  Search
All data on this website is collected from public sources. Our data reflects the most accurate information available at the time of publication.