provided 2a or d is a nonpositive integer.
Corollary 4.11 Let α and β 1 be two nonpositive integers or a be a nonpositive integer.
Let and be real constants such that and neither nor is a nonpositive integer.
Also, we use the convention that if t is a nonpositive integer, but (t+r) is not a nonpositive integer, then (t^{overline{r}}:=0).
Corollary 4.2 Let b 1 and b 2 be two nonpositive integers or α be a nonpositive integer and let c, β ≠ 0, − 1, − 2, … .
Theorem 4.1 Let b 1 and b 2 be two nonpositive integers or α be a nonpositive integer and let c, β ≠ 0, − 1, − 2, … .
We know that if neither nor is a nonpositive integer, then the power series for converges for all values of.
(4) In the case α is a nonpositive integer, the q-factorial function is defined by (t-s)_{q}^{alpha}=t^{alpha}prod _{i=0}^{infty}frac{1- frac{s}{t} q^{i}}{ 1- frac{s}{t} q^{i+alpha}}.
If we set x = 1 in Theorem 4.8, we obtain the following corollary which has been given by Wei et al. [[14], p.8]. Corollary 4.9 Let e be a nonpositive integer.
Similarly to [14], we consider the following the multilinear commutator (H_{Phi,beta,vec{b}}) generalized by the n dimensional fractional Hausdorff operator (H_{Phi,beta}) and b⃗: H_{Phi,beta,vec{b}}f(x)= int_{mathbb{R}^{n}} Biggl[prod_{j=1}^{m} bigl(b_{j}(x -b_{j}(y) bigr) Biggr]frac{Phi(frac {x -b_{j)} yy|^{n-bigr}}f(y), dy, where m is a nonpositive integer, Φ is a radial function.
