Pochhammer k-symbol

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

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

${\displaystyle {\begin{aligned}(x)_{n,k}&=x(x+k)(x+2k)\cdots (x+(n-1)k)=\prod _{i=1}^{n}(x+(i-1)k)\\&=k^{n}\times \left({\frac {x}{k}}\right)_{n},\,\end{aligned}}}$

and the k-gamma function Γ_k, with k > 0, is defined as

${\displaystyle \Gamma _{k}(x)=\lim _{n\to \infty }{\frac {n!k^{n}(nk)^{x/k-1}}{(x)_{n,k}}}.}$

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.

Continued Fractions, Congruences, and Finite Difference Equations

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 ^[2] in the form of the next infinite continued fraction expansion given by

${\displaystyle {\begin{aligned}{\text{Conv}}_{h}(\alpha ,R;z)&:={\cfrac {1}{1-R\cdot z-{\cfrac {\alpha R\cdot z^{2}}{1-(R+2\alpha )\cdot z-{\cfrac {2\alpha (R+\alpha )\cdot z^{2}}{1-(R+4\alpha )\cdot z-{\cfrac {3\alpha (R+2\alpha )\cdot z^{2}}{\cdots }}}}}}}}.\end{aligned}}}$

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

${\displaystyle {\begin{aligned}{\text{Conv}}_{h}(\alpha ,R;z)&:={\cfrac {1}{1-R\cdot z-{\cfrac {\alpha R\cdot z^{2}}{1-(R+2\alpha )\cdot z-{\cfrac {2\alpha (R+\alpha )\cdot z^{2}}{1-(R+4\alpha )\cdot z-{\cfrac {3\alpha (R+2\alpha )\cdot z^{2}}{\cfrac {\cdots }{1-(R+2(h-1)\alpha )\cdot z}}}}}}}}}\\&={\frac {{\text{FP}}_{h}(\alpha ,R;z)}{{\text{FQ}}_{h}(\alpha ,R;z)}}=\sum _{n=0}^{2h-1}p_{n}(\alpha ,R)z^{n}+\sum _{n=2h}^{\infty }{\widetilde {e}}_{h,n}(\alpha ,R)z^{n},\end{aligned}}}$

where the component convergent function sequences, ${\displaystyle {\text{FP}}_{h}(\alpha ,R;z)}$ and ${\displaystyle {\text{FQ}}_{h}(\alpha ,R;z)}$, are given as closed-form sums in terms of the ordinary Pochhammer symbol and the Laguerre polynomials by

${\displaystyle {\begin{aligned}{\text{FP}}_{h}(\alpha ,R;z)&=\sum _{n=0}^{h-1}\left[\sum _{i=0}^{n}{\binom {h}{i}}(1-h-R/\alpha )_{i}(R/\alpha )_{n-i}\right](\alpha z)^{n}\\{\text{FQ}}_{h}(\alpha ,R;z)&=\sum _{i=0}^{h}{\binom {h}{i}}(R/\alpha +h-i)_{i}(-\alpha z)^{i}\\&=(-\alpha z)^{h}\cdot h!\cdot L_{h}^{(R/\alpha -1)}\left((\alpha z)^{-1}\right).\end{aligned}}}$

The rationality of the ${\displaystyle h^{th}}$ convergent functions for all ${\displaystyle h\geq 2}$, combined with known enumerative properties of the J-fraction expansions, imply the following finite difference equations both exactly generating ${\displaystyle (x)_{n,\alpha }}$ for all ${\displaystyle n\geq 1}$, and generating the symbol modulo ${\displaystyle h\alpha ^{t}}$ for some fixed integer ${\displaystyle 0\leq t\leq h}$:

${\displaystyle {\begin{aligned}(x)_{n,\alpha }&=\sum _{0\leq k<n}{\binom {n}{k+1}}(-1)^{k}(x+(n-1)\alpha )_{k+1,-\alpha }(x)_{n-1-k,\alpha }\\(x)_{n,\alpha }&\equiv \sum _{0\leq k\leq n}{\binom {h}{k}}\alpha ^{n+(t+1)k}(1-h-x/\alpha )_{k}(x/\alpha )_{n-k}&&{\pmod {h\alpha ^{t}}}.\end{aligned}}}$

The rationality of ${\displaystyle {\text{Conv}}_{h}(\alpha ,R;z)}$ also implies the next exact expansions of these products given by

${\displaystyle (x)_{n,\alpha }=\sum _{j=1}^{h}c_{h,j}(\alpha ,x)\times \ell _{h,j}(\alpha ,x)^{n},}$

where the formula is expanded in terms of the special zeros of the Laguerre polynomials, or equivalently, of the confluent hypergeometric function, defined as the finite (ordered) set

${\displaystyle \left(\ell _{h,j}(\alpha ,x)\right)_{j=1}^{h}=\left\{z_{j}:\alpha ^{h}\times U\left(-h,{\frac {x}{\alpha }},{\frac {z}{\alpha }}\right)=0,\ 1\leq j\leq h\right\},}$

and where ${\displaystyle {\text{Conv}}_{h}(\alpha ,R;z):=\sum _{j=1}^{h}c_{h,j}(\alpha ,x)/(1-\ell _{h,j}(\alpha ,x))}$ denotes the partial fraction decomposition of the rational ${\displaystyle h^{th}}$ convergent function.

Additionally, since the denominator convergent functions, ${\displaystyle {\text{FQ}}_{h}(\alpha ,R;z)}$, are expanded exactly through the Laguerre polynomials as above, we can exactly generate the Pochhammer k-symbol as the series coefficients

${\displaystyle (x)_{n,\alpha }=\alpha ^{n}\cdot [w^{n}]\left(\sum _{i=0}^{n+n_{0}-1}{\binom {{\frac {x}{\alpha }}+i-1}{i}}\times {\frac {(-1/w)}{(i+1)L_{i}^{(x/\alpha -1)}(1/w)L_{i+1}^{(x/\alpha -1)}(1/w)}}\right),}$

for any prescribed integer ${\displaystyle n_{0}\geq 0}$.

Special Cases

Special cases of the Pochhammer k-symbol, ${\displaystyle (x)_{n,k}}$, 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 ${\displaystyle \alpha }$-factorial functions studied in the last two references by Schmidt:

• The Pochhammer symbol, or rising factorial function: ${\displaystyle (x)_{n,1}\equiv (x)_{n}}$
• The falling factorial function: ${\displaystyle (x)_{n,-1}\equiv x^{\underline {n}}}$
• The single factorial function: ${\displaystyle n!=(1)_{n,1}=(n)_{n,-1}}$
• The double factorial function: ${\displaystyle (2n-1)!!=(1)_{n,2}=(2n-1)_{n,-2}}$
• The multifactorial functions defined recursively by ${\displaystyle n!_{(\alpha )}=n\cdot (n-\alpha )!_{(\alpha )}}$ for ${\displaystyle \alpha \in \mathbb {Z} ^{+}}$ and some offset ${\displaystyle 0\leq d<\alpha }$: ${\displaystyle (\alpha n-d)!_{(\alpha )}=(\alpha -d)_{n,\alpha }=(\alpha n-d)_{n,-\alpha }}$ and ${\displaystyle n!_{(\alpha )}=(n)_{\lfloor (n+\alpha -1)/\alpha \rfloor ,-\alpha }}$

The expansions of these k-symbol-related products considered termwise with respect to the coefficients of the powers of ${\displaystyle x^{k}}$ (${\displaystyle 1\leq k\leq n}$) for each finite ${\displaystyle n\geq 1}$ are defined in the article on generalized Stirling numbers of the first kind and generalized Stirling (convolution) polynomials in.^[3]

References