Pochhammer k-symbol explained

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

\begin{align} (x) _{n,k} & = x(x + k)(x + 2 k) … (x + (n -1 )k) = \prod ^{n} _{i =1} (x + (i -1 )k) \ & = k ^{n} x \left( (x)/(k) \right) _{n,} \end{align}

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

\Gamma _{k(x)} = \lim _{n\toinfty} (n!k ^{n} (nk) ^{x/k})/((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

p _{n(\alpha,} R) : = R(R + \alpha) … (R + (n -1 )\alpha) for fixed \alpha > 0 and some indeterminate parameter R , are considered in in the form of the next infinite continued fraction expansion given by\begin{align} Conv _{h(\alpha,} R; z) & : = \cfrac{ 1 }{ 1 - R ⋅ z - \cfrac{\alpha R ⋅ z ^{2}{} 1 - (R +2 \alpha) ⋅ z - \cfrac{ 2 \alpha (R + \alpha) ⋅ z ^{2}{} 1 - (R + 4 \alpha) ⋅ z - \cfrac{ 3 \alpha (R + 2 \alpha) ⋅ z ^{2}{} … }}}}. \end{align}

The rational

h ^{th} convergent function, Conv _{h(\alpha,} R; z) , to the full generating function for these products expanded by the last equation is given by\begin{align} Conv _{h(\alpha,} R; z) & : = \cfrac{ 1 }{ 1 - R ⋅ z - \cfrac{\alpha R ⋅ z ^{2}{} 1 - (R +2 \alpha) ⋅ z - \cfrac{ 2 \alpha (R + \alpha) ⋅ z ^{2}{} 1 - (R + 4 \alpha) ⋅ z - \cfrac{ 3 \alpha (R + 2 \alpha) ⋅ z ^{2}{} \cfrac{ … }{ 1 - (R + 2 (h -1 ) \alpha) ⋅ z}}}}} \ & = (FP _{h(\alpha,} R; z))/(FQ _{h(\alpha,} R; z)) = \sum ^{2 h -1} _{n =0} p _{n(\alpha,} R) z ^{n} + \sum ^{inf ty} _{n =2 h} \widetilde{e} _{h,n} (\alpha, R) z ^{n,} \end{align}

where the component convergent function sequences,

FP _{h(\alpha,} R; z) and FQ _{h(\alpha,} R; z) , are given as closed-form sums in terms of the ordinary Pochhammer symbol and the Laguerre polynomials by\begin{align} FP _{h(\alpha,} R; z) & = \sum ^{h -1} _{n =0} \left[\sum ^{n} _{i =0} \binom{h}{i} ( 1- h - R/\alpha) _{i} (R/\alpha) _{n-i} \right] (\alpha z) ^{n} \ FQ _{h(\alpha,} R; z) & = \sum ^{h} _{i =0} \binom{h}{i} (R/\alpha + h - i) _{i(-\alpha} z) ^{i} \ & = ( - \alpha z) ^{h} ⋅ h! ⋅ L ^{(R/\alpha -1 )} _{h} \left((\alpha z) ^{-1} \right). \end{align}

The rationality of the

h ^{th} convergent functions for all h \geq 2 , combined with known enumerative properties of the J-fraction expansions, imply the following finite difference equations both exactly generating (x) _{n,\alpha} for all n \geq 1 , and generating the symbol modulo h \alpha ^{t} for some fixed integer 0 \leq t \leq h :\begin{align} (x) _{n,\alpha} & = \sum _{0} \binom{n}{k +1 } ( -1 ) ^{k} (x + (n -1 )\alpha) _{k+1,-\alpha} (x) _{n-1-k,\alpha} \ (x) _{n,\alpha} & \equiv \sum _{0} \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

Conv _{h(\alpha,} R; z) also implies the next exact expansions of these products given by(x) _{n,\alpha} = \sum ^{h} _{j =1} c _{h,j} (\alpha, x) x \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

\left(\ell _{h,j} (\alpha, x)\right) ^{h} _{j =1} = \left\{ z _{j} : \alpha ^{h} x U\left( - h, (x)/(\alpha) , (z)/(\alpha) \right) = 0 , 1 \leq j \leq h \right\},

and where

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

Additionally, since the denominator convergent functions,

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(x) _{n,\alpha} = \alpha ^{n} ⋅ [w ^{n + n _{0-1}} _{i =0} \binom{ (x)/(\alpha) + i -1 }{i} x (( -1 /w))/((i +1 ) L ^{(x/\alpha -1 )} _{i} ( 1 /w) L ^{(x/\alpha -1 )} _{i +1} ( 1 /w)) \right),

for any prescribed integer

n _{0} \geq 0 .

Special Cases

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 \alpha -factorial functions studied in the last two references by Schmidt:

• The Pochhammer symbol, or rising factorial function:

(x) _{n,1} \equiv (x) _{n}

• The falling factorial function:

(x) _{n,-1} \equiv x ^{\underline{n} }

• The single factorial function:

n! = ( 1 ) _{n,1} = (n) _{n,-1}

• The double factorial function:

( 2 n -1 )!! = ( 1 ) _{n,2} = ( 2 n -1 ) _{n,-2}

• The multifactorial functions defined recursively by

n! _{(\alpha)} = n ⋅ (n - \alpha)! _{(\alpha)} for \alpha \in Z ^{+} and some offset 0 \leq d < \alpha : (\alpha n - d)! _{(\alpha)} = (\alpha - d) _{n,\alpha} = (\alpha n - d) _{n,-\alpha} and n! _{(\alpha)} = (n) _{\lfloor}

The expansions of these k-symbol-related products considered termwise with respect to the coefficients of the powers of

x ^{k} (1 \leq k \leq n ) for each finite n \geq 1 are defined in the article on generalized Stirling numbers of the first kind and generalized Stirling (convolution) polynomials in.

References

1. 2005 . math/0405596 . Rafael . Díaz . Eddy Pariguan . On hypergeometric functions and k-Pochhammer symbol .