We generalize the notions of the fractional q-integral and q-derivative by introducing variable lower limit of integration. We discuss some properties and their relations. Finally, we give a q-Taylor-like formula which includes fractional q-derivatives of the function.

Abstract. In this paper, we define a new class of almost orthogonal polynomials which can be used successfully for modelling of electronic systems which generate orthonormal basis. Especially, for the classical weight function, they can be considered like a generalization of the classical orthogonal polynomials (Legendre, Laguerre, Hermite, . . . ). They are very suitable for analysis and synthesis of imperfect technical systems which are projected to generate orthogonal polynomials, but in the reality generate almost orthogonal polynomials.

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PurposePterygium is a common lesion affecting the population in countries with high levels of ultraviolet exposure. The final shape of a pterygium is the result of a growth pattern, which remains poorly understood. This manuscript provides a mathematical analysis as a tool to determine the shape of human pterygia.Materials and methodsEighteen patients, all affected by nasal unilateral pterygia, were randomly selected from our patient database independently of sex, origin, or race. We included all primary or recurrent pterygia with signs of proliferation, dry eye, and induction of astigmatism. Pseudopterygia were excluded from this study. Pterygia were outlined and analyzed mathematically using a Cartesian coordinate system with two axes (X, Y) and five accurate landmarks of the pterygium.ResultsIn 13 patients (72%), the shape of the pterygia was hyperbolic and in five patients (28%), the shape was rather elliptical.ConclusionThis analysis gives a highly accurate mathematical description of the shape of human pterygia. This might help to better assess the clinical results and outcome of the great variety of therapeutic approaches concerning these lesions.