| Summary: | Generalized Laguerre polynomials, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>L</mi><mi>n</mi><mrow><mo>(</mo><mi>α</mi><mo>)</mo></mrow></msubsup></semantics></math></inline-formula>, verify the well-known Rodrigues’ formula. Using Weyl and Riemann–Liouville fractional calculi, we present several fractional generalizations of Rodrigues’ formula for generalized Laguerre functions and polynomials. As a consequence, we give a new addition formula and an integral representation for these polynomials. Finally, we introduce a new family of fractional Lebesgue spaces and show that some of these special functions belong to them.
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