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Fractional calculus of the generalized Wright function. (English) Zbl 1144.26008

For the generalized Wright function defined by \[ ._p\Psi_q(z)=\sum\limits_{k=0}^\infty\frac{\prod_{i=1}^p\Gamma(a_i+\alpha_i k)z^k} {\prod_{j=1}^q\Gamma(b_j+\beta_j k)}k! \]
there are proved various formulas for its Riemann-Liouville fractional integrals and derivatives. One of the formulas (Theorem 1) states that
\[ I_{0+}^\alpha (t^{\gamma-1}_p\Psi_q\left(at^\mu\right))(x)=x^{\gamma+\alpha-1} \left._{p+1}\Psi_{q+1}\left(ax^\mu\right)\right. \]
with \(a_{p+1}=\gamma, \alpha_{p+1}=\mu, b_{p+1}=\gamma+\alpha, \beta_{p+1}=\mu,\) under the assumptions that \( a\in \mathbb{C}, \alpha\in \mathbb{C}, \operatorname{Re}\gamma>0, \mu>0\).
Similar formulas for the left-hand side fractional integration \(I_-^\alpha\) and for fractional derivatives are also proved. Some particular cases are also separately given (the cases of the usual Wright function and Bessel-Maitland function).

MSC:

26A33 Fractional derivatives and integrals
33C20 Generalized hypergeometric series, \({}_pF_q\)
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