Generalized Hypergeometric function-Definition, Convergence of the series for , Differential equation and its solution, Contiguous function relations, Saalschutz’s theorem, whipple’s theorem. Contour integral representation for , Eulerian type integrals involving , Integral representation for .
Unit II:
II
Meijer’s G function- Definition, Elementary properties, Multiplication formulas, Derivatives, Mellin and Laplace transforms of the G- function.
Unit III:
III
H-function of one variable: Definition, Identities, Special cases, Differentiation formulas, Recurrence and contiguous function relations, Finite and infinite series, Fourier series for the H-function, Simple finite and infinite integrals involving the H-function.
Unit IV:
IV
Fractional Calculus: Definition and elementary properties of Riemann-Liouville fractional integral and derivatives, Derivative of the fractional integral, The fractional integral of derivatives.
Unit V:
V
Leibnitz’s formula for fractional integral and fractional derivatives, Law of exponent, Image of elementary and generalized hypergeometric function under fractional integrals and derivatives.
Essential Readings:
W.N. Bailey, Generalized Hyper Geometric Series, Cambridge tracts in Mathematical and Mathematical Physics No. 32, Cambridge University Press, 1935.
A.M. Mathai, R.K. Saxena, Generalized Hyper Geometric Functions with Applications in Statistic and Physical Sciences, Springer- Verlag, Lecture Notes Series No. 348, Heidelberg and New York, 1973.
A.M. Mathai, R.K.Saxena,The H-Function with Applications in Statistic and Other Disciplines, John Wiley and sons, New York,1978.
E. D. Rainville, Special Functions, Macmillan, New York,1960.
I.N. Sneddon, Special Functions of Mathematical Physics and Chemistry, 2nd edition, Oliver and Boyd Edinburgh, 1961..
Kenneth Miller, Bertram Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley-Interscience, 1 edition, 1993.
Rudolf Hilfer, Applications of Fractional Calculus in Physics, World Scientific, 2000.
