Funzione q-esponenziale

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Nella matematica combinatoria e nello studio delle funzioni speciali il termine q-esponenziale viene usato per due q-analoghi della classica funzione esponenziale.

Definizioni

Consideriamo le seguenti funzioni

e_{q}(z):=\sum _{n=0}^{\infty }{\frac {z^{n}}{(q;q)_{n}}}=\prod _{n=0}^{\infty }(1-zq^{n})^{-1}={\frac {1}{(z;q)_{\infty }}}

E_{q}(z):=\sum _{n=0}^{\infty }{\frac {q^{n \choose 2}z^{n}}{(q;q)_{n}}}=\prod _{n=0}^{\infty }(1+q^{n}z)=(-z;q)_{\infty }

dove

(z;q)_{n}:=(1-z)(1-zq)\cdots (1-zq^{n-1})

è il q-fattoriale crescente. Che la prima funzione costituisca un q-analogo dell'esponenziale ordinario segue dalla proprietà

\left({\frac {d}{dz}}\right)_{q}e_{q}(z)=e_{q}(z)

dove l'operatore di derivazione a sinistra è la q-derivata. L'identità precedente si verifica facilmente considerando la q-derivata del monomio

\left({\frac {d}{dz}}\right)_{q}z^{n}=z^{n-1}{\frac {1-q^{n}}{1-q}}=[n]_{q}z^{n-1}

Qui $[n]_{q}$ denota il q-bracket.

Proprietà

Per q reale con $q<1$ la funzione $e_{q}(z)$ è una funzione intera di z.

Espressione ipergeometrica

In termini della q-serie ipergeometrica, la prima funzione q-esponenziale $e_{q}(t)$ viene espressa da

e_{q}(z)=\;_{1}\phi _{0}(0;q,z)

Esiste una simile espressione per la seconda funzione in termini della q-serie ipergeometrica generalizzata.

Collegamenti esterni

(EN) Eric W. Weisstein, Funzione q-esponenziale, su MathWorld, Wolfram Research.

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