Theory of generalized trigonometric functions: from Laguerre to Airy forms. (English) Zbl 1407.33006
Summary: We develop a new point of view to introduce families of functions, which can be identified as generalization of the ordinary trigonometric or hyperbolic functions. They are defined using a procedure based on umbral methods, inspired by the Bessel calculus of Bochner, Cholewinsky and Haimo. We propose further extensions of the method and of the relevant concepts as well and obtain new families of integral transforms allowing the framing of the previous concepts within the context of generalized Borel transform.
MSC:
| 33C10 | Bessel and Airy functions, cylinder functions, \({}_0F_1\) |
| 33B10 | Exponential and trigonometric functions |
Keywords:
trigonometric and hyperbolic functions; Laguerre polynomials; Airy forms; umbral calculus; integral transforms; operational methodsReferences:
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