In the mathematical theory of special functions, the Pochhammer k-symbol and the k-gamma function, introduced by Rafael Díaz and Eddy Pariguan [1] are generalizations of the Pochhammer symbol and gamma function. They differ from the Pochhammer symbol and gamma function in that they can be related to a general arithmetic progression in the same manner as those are related to the sequence of consecutive integers.
Definition
edit
The Pochhammer k-symbol (x)n,k is defined as
and the k-gamma function Γk, with k > 0, is defined as
When k = 1 the standard Pochhammer symbol and gamma function are obtained.
Díaz and Pariguan use these definitions to demonstrate a number of properties of the hypergeometric function. Although Díaz and Pariguan restrict these symbols to k > 0, the Pochhammer k-symbol as they define it is well-defined for all real k, and for negative k gives the falling factorial, while for k = 0 it reduces to the powerxn.
The Díaz and Pariguan paper does not address the many analogies between the Pochhammer k-symbol and the power function, such as the fact that the binomial theorem can be extended to Pochhammer k-symbols. It is true, however, that many equations involving the power function xn continue to hold when xn is replaced by (x)n,k.
Continued Fractions, Congruences, and Finite Difference Equations
edit
Jacobi-type J-fractions for the ordinary generating function of the Pochhammer k-symbol, denoted in slightly different notation by for fixed and some indeterminate parameter , are considered in [2]
in the form of the next infinite continued fraction expansion given by
The rational convergent function, , to the full generating function for these products expanded by the last equation is given by
where the component convergent function sequences, and , are given as closed-form sums in terms of the ordinary Pochhammer symbol and the Laguerre polynomials by
The rationality of the convergent functions for all , combined with known enumerative properties of the J-fraction expansions, imply the following finite difference equations both exactly generating for all , and generating the symbol modulo for some fixed integer :
The rationality of also implies the next exact expansions of these products given by
Additionally, since the denominator convergent functions, , are expanded exactly through the Laguerre polynomials as above, we can exactly generate the Pochhammer k-symbol as the series coefficients
for any prescribed integer .
Special Cases
edit
Special cases of the Pochhammer k-symbol, , correspond to the following special cases of the falling and rising factorials, including the Pochhammer symbol, and the generalized cases of the multiple factorial functions (multifactorial functions), or the -factorial functions studied in the last two references by Schmidt:
^
Schmidt, Maxie D. (2017), Jacobi-Type Continued Fractions for the Ordinary Generating Functions of Generalized Factorial Functions, vol. 20, J. Integer Seq., arXiv:1610.09691
^
Schmidt, Maxie D. (2010), Generalized j-Factorial Functions, Polynomials, and Applications, vol. 13, J. Integer Seq.