Many mathematical entities and objects are attributed to (and named after) him. These entities and objects include (among other items) Srivastava’s polynomials and functions, Carlitz-Srivastava polynomials, Srivastava-Buschman polynomials, Srivastava-Singhal polynomials, Chan-Chyan-Srivastava polynomials, Erkuş-Srivastava polynomials, Srivastava-Daoust multivariable hypergeometric function, Srivastava-Panda multivariable H-function, Singhal-Srivastava generating function, Srivastava-Agarwal basic (or q-) generating function, and Wu-Srivastava inequality in the field of higher transcendental functions; Srivastava-Owa, Choi-Saigo-Srivastava, Jung-Kim-Srivastava, Liu-Srivastava, Cho-Kwon-Srivastava, Dziok-Srivastava, Srivastava-Attiya and Srivastava-Wright operators in the field of geometric function theory in complex analysis; Srivastava-Gupta operator in the field of approximation theory; the Srivastava, Adamchik-Srivastava and Choi-Srivastava methods in the field of analytic number theory; and so on.
Professor Srivastava has supervised (and is currently supervising) a number of post-graduate students working toward their Master’s, Ph. D. and/or D. Sc. degrees in different parts of the world. Besides, many post-doctoral fellows and research associates have worked with him at West Virginia University in the USA and at the University of Victoria in Canada.
Some of the significant and remarkable contributions by Professor Srivastava are listed below under each of the main topics of his current research interests.
(i) Real and complex analysis: A unified theory of numerous potentially useful function classes, and of various integral and convolution operators using hypergeometric functions, especially in geometric function theory in complex analysis, and several classes of analytic and geometric inequalities in the field of real analysis.
(ii) Fractional calculus and its applications: Generalizations of such classical fractional-calculus operators as the Riemann-Liouville and Weyl operators together with their fruitful applications to numerous families of differential, integral, and integro-differential equations, especially some general classes of fractional kinetic equations, and also to some Volterra-type integro-differential equations which emerge from the unsaturated behavior of the free electron laser.
(iii) Integral equations and transforms: Explicit solutions of several general families of dual series and integral equations occurring in potential theory; unified theory of many known generalizations of the classical Laplace transform (such as the Meijer and Varma transforms) and of other multiple integral transforms by means of the Whittaker -function and the (Srivastava-Panda) multivariable H-function in their kernels.
(iv) Higher transcendental functions and their applications: Discovery, introduction, and systematic (and unified) investigation of a set of 205 triple Gaussian hypergeometric series, especially the triple hypergeometric functions , , and , added to the 14-member set conjectured and defined in 1893 by Giuseppe Lauricella (1867-1913). Unified theory and applications of the multivariable extensions of the celebrated higher transcendental (Ψ- and H-) functions of Charles Fox (1897-1977) and Edward Maitland Wright (1906-2005), and also of the Mittag-Leffler E-functions named after Gustav Mittag-Leffler (1846-1927). Mention should be made also of his applications of some of these higher transcendental functions in quantum and fluid mechanics, astrophysics, probability distribution theory, queuing theory and other related stochastic processes, and so on.
(v) q-Series and q-polynomials: Basic theory of general q-polynomial expansions for functions of several complex variables, extensions of several celebrated q-identities of Srinivasa Ramanujan (1887-1920), and systematic introduction and investigation of multivariable basic (or q-) hypergeometric series.
(vi) Analytic number theory: Presentation of several computationally-friendly and rapidly-converging series representations for Riemann’s zeta function, Dirichlet’s L-series, introduction and application of some novel techniques for closed-form evaluations of series involving a wide variety of sequences and functions of analytic number theory, and so on. His applications of (especially) the Hurwitz-Lerch zeta function in geometric function theory in complex analysis and in probability distribution theory and related topics of statistical sciences deserve to be recorded here.
Professor Srivastava’s publications have been reviewed by (among others) Mathematical Reviews (USA), Referativnyi Zhurnal Matematika (Russia), Zentralblatt für Mathematik (Germany), and Applied Mechanics Reviews (USA) under various 2010 Mathematical Subject Classifications (MathSciNet) including (for example) the following general classifications:
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01
History and Biography
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05
Combinatorics
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11
Number Theory
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15
Linear and Multilinear Algebra; Matrix Theory
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26
Real Functions
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30
Functions of a Complex Variable
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31
Potential Theory
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33
Special Functions
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34
Ordinary Differential Equations
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35
Partial Differential Equations
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39
Difference and Functional Equations
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40
Sequences, Series, Summability
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41
Approximations and Expansions
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42
Fourier Analysis
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44
Integral Transforms, Operational Calculus
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45
Integral Equations
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46
Functional Analysis
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47
Operator Theory
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58
General Global Analysis, Analysis on Manifolds
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60
Probability Theory and Stochastic Processes
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62
Statistics
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76
Fluid Mechanics
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78
Optics, Electromagnetic Theory
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81
Quantum Theory
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85
Astronomy and Astrophysics
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90
Operations Research, Mathematical Programming
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91
Game Theory, Economics, Social and Behavioral Sciences
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93
Systems Theory, Control