Hypergeometric functions

After substitution of (A3.1) into (6.14), several integrals of the same type must be calculated. These integrals can be expressed via the degenerate hypergeometrical function d>(-, -, -) and gamma-function T( ) ... 

Although the hypergeometric functions are useful in spectroscopy, as they describe the rotation of a symmetric top molecule (Section 9.2.4), their importance is primarily due to their generality. If, for example, a = 1 and fi say, Eq. (154) becomes a +i — a for all values of n. The result is the ordinary geometric series... 

The Chebyshev polynomials, whiGh occur in quantum chemistry and in certain numerical applications, can be obtained from the hypergeometric functions by placing a = -/ , an integer, and y — Finally, the hypergeometric... 

A sequence of approximations, using properties of the confluent hypergeometric function, integration by steepest descents, and judicious discard of all but the dominant terms, gives one the asymptotic form... 

By direct integration or by specializing overlap coefficients between alternative harmonics [25] we are able to write it directly as a single sum of the Racah type. This sum [26] is a hypergeometric function 4F3 of unit argument ... 

The non-linear theory of steady-steady (quasi-steady-state/pseudo-steady-state) kinetics of complex catalytic reactions is developed. It is illustrated in detail by the example of the single-route reversible catalytic reaction. The theoretical framework is based on the concept of the kinetic polynomial which has been proposed by authors in 1980-1990s and recent results of the algebraic theory, i.e. an approach of hypergeometric functions introduced by Gel fand, Kapranov and Zelevinsky (1994) and more developed recently by Sturnfels (2000) and Passare and Tsikh (2004). The concept of ensemble of equilibrium subsystems introduced in our earlier papers (see in detail Lazman and Yablonskii, 1991) was used as a physico-chemical and mathematical tool, which generalizes the well-known concept of equilibrium step . In each equilibrium subsystem, (n—1) steps are considered to be under equilibrium conditions and one step is limiting n is a number of steps of the complex reaction). It was shown that all solutions of these equilibrium subsystems define coefficients of the kinetic polynomial. 

Klein considered this approach too cumbersome. Note that hypergeometric functions were applied to problem (54) as early as in 18th century. 

Note that Eq. (126) implies a nonzero initial velocity of the free boundary, in common with previous exact solutions, which were, however, selfsimilar. The present problem, while linear, is still in the form of a partial differential equation. However, it is readily solved by separation of variables, leading to an ordinary differential equation of the confluent hypergeometric form. The solution appears in terms of the confluent hypergeometric function of the first kind, defined by... 

To these three a priori reasons for considering the generalization to two indices a fourth may be added a posteriori. We shall encounter, in important specific cases, one or two of the rarer special functions associated with the confluent hypergeometric function. 

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