论文标题
$α$，$β$ - 相关子组的Riordan矩阵的扩张
$α$, $β$-expansions of the Riordan matrices of the associated subgroup
论文作者
论文摘要
我们考虑矩阵$ \ left的组（1，g \ left（x \ right）\ right）$ isomorphic to正式功率系列$ g \ left（x \ right）= x+{x+{{g} _ {2} _ {2}}}}}}} 1，{{g} _ {2}} \ left（x \ right）\右） \ right）\ right）$。表示$ p_ {k}^{α} = \ left（1，x {{{\ left（1-kα{{x}^{k}}}}} \ right）}}^{ - { - 1}/{k}/{k} \;}}}}}}}} \ right）$。矩阵$ \ left（1，g \ left（x \ right）\右）$被分解为矩阵$ p_ {k}^{α} $的无限产品美元\ right）= ... p_ {k}^{{{{α} _ {k}}} ... p_ {2}^{{{{α} _ {2}}}} p_ {1}} }}} = p_ {1}^{{{{β} _ {1}}} p_ {2}^{{{β} {{β} _ {2}}}}}}} ...我们获得两个表示系列系数$ {{{\ left（{g \ left（x \ oright）}/{x} \; \ right）}^{z}} $的系列的系数，以扩展系数$ {{α}} _ {i}} $，$ i} $ {i} $，$，$ {系列$ g_ {α}^{\ left（t \ right）} \ left（x \ right）$和$ g_ {β}^{\ left（t \ right）} \ left（x \ orirt）$的单参数系列。
We consider the group of the matrices $\left( 1,g\left( x \right) \right)$ isomorphic to the group of formal power series $g\left( x \right)=x+{{g}_{2}}{{x}^{2}}+...$ under composition: $\left( 1,{{g}_{2}}\left( x \right) \right)\left( 1,{{g}_{1}}\left( x \right) \right)=\left( 1,{{g}_{1}}\left( {{g}_{2}}\left( x \right) \right) \right)$. Denote $P_{k}^{α}=\left( 1,x{{\left( 1-kα{{x}^{k}} \right)}^{{-1}/{k}\;}} \right)$. Matrix $\left( 1,g\left( x \right) \right)$is decomposed into an infinite product of the matrices $P_{k}^{α}$ with suitable exponents in two ways: to left-handed and right-handed products with respect to the matrix $P_{1}^{{{α}_{1}}={{β}_{1}}}$: $\left( 1,g\left( x \right) \right)=...P_{k}^{{{α}_{k}}}...P_{2}^{{{α}_{2}}}P_{1}^{{{α}_{1}}}=P_{1}^{{{β}_{1}}}P_{2}^{{{β}_{2}}}...P_{k}^{{{β}_{k}}}...$. We obtain two formulas expressing the coefficients of the series ${{\left( {g\left( x \right)}/{x}\; \right)}^{z}}$ in terms of the expansion coefficients ${{α}_{i}}$, ${{β}_{i}}$ and introduce two one-parameter families of series $g_{α}^{\left( t \right)}\left( x \right)$ and $g_{β}^{\left( t \right)}\left( x \right)$ associated with these expansions.
