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A generalization of the falling factorial in which a function is evaluated on a descending arithmetic sequence of integers and the values are multiplied is:
:<mathtex>[f(x)]^{k/-h}=f(x)\cdot f(x-h)\cdot f(x-2h)\cdots f(x-(k-1)h),</mathtex>
where {{math|−''h''}} is the decrement and {{math|''k''}} is the number of factors. The corresponding generalization of the rising factorial is
:<mathtex>[f(x)]^{k/h}=f(x)\cdot f(x+h)\cdot f(x+2h)\cdots f(x+(k-1)h).</mathtex>
This notation unifies the rising and falling factorials, which are [''x'']<sup>''k''/1</sup> and [''x'']<sup>''k''/−1</sup>, respectively.
For any fixed arithmetic function <mathtex>f: \mathbb{N} \rightarrow \mathbb{C}</mathtex> and symbolic parameters <mathtex>x, t</mathtex>, related generalized factorial products of the form
:<mathtex>(x)_{n,f,t} := \prod_{k=1}^{n-1} \left(x+\frac{f(k)}{t^k}\right)</mathtex>
may be studied from the point of view of the classes of generalized [[Stirling numbers of the first kind]] defined by the following coefficients of the powers of <math>x</math> in the expansions of <math>(x)_{n,f,t}</math> and then by the next corresponding triangular recurrence relation:
:<mathtex>\begin{align} \left[\begin{matrix} n \\ k \end{matrix} \right]_{f,t} & = [x^{k-1}] (x)_{n,f,t} \\ & = f(n-1) t^{1-n} \left[\begin{matrix} n-1 \\ k \end{matrix} \right]_{f,t} + \left[\begin{matrix} n-1 \\ k-1 \end{matrix} \right]_{f,t} + \delta_{n,0} \delta_{k,0}. \end{align} </mathtex>
These coefficients satisfy a number of analogous properties to those for the [[Stirling numbers of the first kind]] as well as recurrence relations and functional equations related to the ''f-harmonic numbers'', <mathtex>F_n^{(r)}(t) := \sum_{k \leq n} t^k / f(k)^r</mathtex>.<ref>''[https://arxiv.org/abs/1611.04708 Combinatorial Identities for Generalized Stirling Numbers Expanding f-Factorial Functions and the f-Harmonic Numbers]'' (2016).</ref>
== См.также ==
