Definition:f

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The LaTeX DLMF and DRMF macro \f represents Function.

This macro is in the category of polynomials.

In math mode, this macro can be called in the following ways:

\f{f} produces ${\displaystyle{\displaystyle{\displaystyle{f}}}}$

\f{f}@{x} produces ${\displaystyle{\displaystyle{\displaystyle{f}\!\left(x\right)}}}$

These are defined by

${\displaystyle{\displaystyle S^{(\alpha,\beta)}_{2m}\left(x\right):={\rm const}\times P^{(\alpha,\beta)}_{m}\left(2x^{2}-1\right),}}$

${\displaystyle{\displaystyle S^{(\alpha,\beta)}_{2m+1}\left(x\right):={\rm const}\times x\,P^{(\alpha,\beta+1)}_{m}\left(2x^{2}-1\right).}}$

Then for ${\displaystyle{\displaystyle\alpha,\beta>-1}}$, we have the orthogonality relation

${\displaystyle{\displaystyle\int_{-1}^{1}S^{(\alpha,\beta)}_{m}\left(x\right)\,S^{(\alpha,\beta)}_{n}\left(x\right)\,|x|^{2\beta+1}(1-x^{2})^{\alpha}\,dx=0,}}$

for ${\displaystyle{\displaystyle m\neq n}}$.

Symbols List

${\displaystyle{\displaystyle{\displaystyle{f}}}}$ : function : http://drmf.wmflabs.org/wiki/Definition:f
${\displaystyle{\displaystyle{\displaystyle S^{(\alpha,\beta)}_{n}}}}$ : Generalized Gegenbauer polynomial : http://drmf.wmflabs.org/wiki/Definition:GenGegenbauer
${\displaystyle{\displaystyle{\displaystyle P^{(\alpha,\beta)}_{n}}}}$ : Jacobi polynomial : http://dlmf.nist.gov/18.3#T1.t1.r3
${\displaystyle{\displaystyle{\displaystyle\int}}}$ : integral : http://dlmf.nist.gov/1.4#iv

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