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Results tagged with orthogonal-polynomials
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user 46140
A familly of orthogonal polynomials is a sequence of polynomials in one variable, one in each degree, such that any two of them are orthogonal with respect to some fixed scalar product on the space of polynomials. They are closely related to continued fractions and useful in harmonic analysis. There are many different families of orthogonal polynomials, among which one can cite Hermite polynomials, Laguerre polynomials, and Jacobi polynomials.
11
votes
Accepted
Do you recognize this sequence of polynomials?
Let $b_n(t)$ be the Morgan-Voyce polynomial defined by $$\begin{eqnarray}b_0(t) &=& 1 \\
b_1(t) &=& t + 1 \\
b_n(t) &=& (t+2) b_{n-1}(t) - b_{n-2}(t)
\end{eqnarray}$$
Then $f_n(t) = (-1)^n b_n(-t)$ fi …
- 7,226
3
votes
A special class of weighted Motzkin paths
It's going to be useful to say that horizontal steps at height $0$ have weight $u$ so that we can derive a recurrence: therefore I shall consider $c_n(t, u)$ and look to specialise it to $u=1$ later.
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- 7,226