Pochhammer k-symbol

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

: $$

\begin{align}

(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{align}

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

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

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.

Jacobi-type J-fractions for the ordinary generating function of the Pochhammer k-symbol, denoted in slightly different notation by $p\_n(\backslash alpha,\; R)\; :=\; R(R+\backslash alpha)\backslash cdots(R+(n-1)\backslash alpha)$ for fixed $\backslash alpha\; >\; 0$ and some indeterminate parameter $R$, are considered in

{{citation

|date=2017

|publisher=J. Integer Seq.

|volume=20

|url=https://cs.uwaterloo.ca/journals/JIS/VOL20/Schmidt/schmidt14.html

|first=Maxie D. |last=Schmidt

|title=Jacobi-Type Continued Fractions for the Ordinary Generating Functions of Generalized Factorial Functions

|arxiv=1610.09691

}}

in the form of the next infinite continued fraction expansion given by

:$$

\begin{align}

\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{align}

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

:$$

\begin{align}

\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{align}

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

:$$

\begin{align}

\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{align}

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

:$$

\begin{align}

(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{align}

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

:$(x)\_\{n,\backslash alpha\}\; =\; \backslash sum\_\{j=1\}^h\; c\_\{h,j\}(\backslash alpha,\; x)\; \backslash times\; \backslash ell\_\{h,j\}(\backslash alpha,\; x)^n,$

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

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

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

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

:$(x)\_\{n,\backslash alpha\}\; =\; \backslash alpha^n\; \backslash cdot\; [w^n]\backslash left(\backslash sum\_\{i=0\}^\{n+n\_0-1\}\; \backslash binom\{\backslash frac\{x\}\{\backslash alpha\}+i-1\}\{i\}\; \backslash times\; \backslash frac\{(-1/w)\}\{(i+1)\; L\_i^\{(x/\backslash alpha-1)\}(1/w)\; L\_\{i+1\}^\{(x/\backslash alpha-1)\}(1/w)\}\backslash right),$

for any prescribed integer $n\_0\; \backslash geq\; 0$.

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

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

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

{{citation

|date=2010

|publisher=J. Integer Seq.

|volume=13

|url=https://cs.uwaterloo.ca/journals/JIS/VOL13/Schmidt/multifact.html

|first=Maxie D. |last=Schmidt

|title=Generalized j-Factorial Functions, Polynomials, and Applications

}}

Category:Gamma and related functions

Category:Factorial and binomial topics