In the mathematical theory of special functions, the Pochhammer k-symbol and the k-gamma function, introduced by Rafael Díaz and Eddy Pariguan 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.

The Pochhammer k-symbol (x)_n,k is defined as

and the k-gamma function Γ_k, 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 power xⁿ.

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.

Jacobi-type J-fractions for the ordinary generating function of the Pochhammer k-symbol, denoted in slightly different notation by ${\displaystyle p_{n}(\alpha ,R):=R(R+\alpha )\cdots (R+(n-1)\alpha )}$ for fixed ${\displaystyle \alpha >0}$ and some indeterminate parameter ${\displaystyle R}$, are considered in in the form of the next infinite continued fraction expansion given by

The rational ${\displaystyle h^{th}}$ convergent function, ${\displaystyle {\text{Conv}}_{h}(\alpha ,R;z)}$, to the full generating function for these products expanded by the last equation is given by