Inverse of power series with alternating coefficients
Given a power series $$f(x)=sum_{i=0}^infty A_{i+1}x^{ni+1},$$ where $n$ is a positive integer and the coefficients $A_i$ are alternating ($A_1,A_3,...>0$ and $A_2,A_4,...<0$) it seems that the coefficients $B_i$ for the inverse function $f^{^-1}(y)=sum_{i=0}^infty B_i y^i$ are always positive (or zero), i.e. $f^{-1}$ is absolutely monotonic (assuming convergence of the series). Does anyone know of an (elementary) proof of this assertion? Using the explicit expression for $B_i$ in terms of Bell polynomials does not get me very far, and I have searched the literature on absolutely monotonic functions in vain.
power-series inverse-function
asked Jan 4 at 9:37
asmithasmith
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Given a power series $$f(x)=sum_{i=0}^infty A_{i+1}x^{ni+1},$$ where $n$ is a positive integer and the coefficients $A_i$ are alternating ($A_1,A_3,...>0$ and $A_2,A_4,...<0$) it seems that the coefficients $B_i$ for the inverse function $f^{^-1}(y)=sum_{i=0}^infty B_i y^i$ are always positive (or zero), i.e. $f^{-1}$ is absolutely monotonic (assuming convergence of the series). Does anyone know of an (elementary) proof of this assertion? Using the explicit expression for $B_i$ in terms of Bell polynomials does not get me very far, and I have searched the literature on absolutely monotonic functions in vain.
power-series inverse-function
asked Jan 4 at 9:37
asmithasmith
11
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asmith is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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Given a power series $$f(x)=sum_{i=0}^infty A_{i+1}x^{ni+1},$$ where $n$ is a positive integer and the coefficients $A_i$ are alternating ($A_1,A_3,...>0$ and $A_2,A_4,...<0$) it seems that the coefficients $B_i$ for the inverse function $f^{^-1}(y)=sum_{i=0}^infty B_i y^i$ are always positive (or zero), i.e. $f^{-1}$ is absolutely monotonic (assuming convergence of the series). Does anyone know of an (elementary) proof of this assertion? Using the explicit expression for $B_i$ in terms of Bell polynomials does not get me very far, and I have searched the literature on absolutely monotonic functions in vain.
power-series inverse-function
asked Jan 4 at 9:37
asmithasmith
11
New contributor
asmith is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Given a power series $$f(x)=sum_{i=0}^infty A_{i+1}x^{ni+1},$$ where $n$ is a positive integer and the coefficients $A_i$ are alternating ($A_1,A_3,...>0$ and $A_2,A_4,...<0$) it seems that the coefficients $B_i$ for the inverse function $f^{^-1}(y)=sum_{i=0}^infty B_i y^i$ are always positive (or zero), i.e. $f^{-1}$ is absolutely monotonic (assuming convergence of the series). Does anyone know of an (elementary) proof of this assertion? Using the explicit expression for $B_i$ in terms of Bell polynomials does not get me very far, and I have searched the literature on absolutely monotonic functions in vain.
power-series inverse-function
power-series inverse-function
asked Jan 4 at 9:37
asmithasmith
11
New contributor
asmith is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked Jan 4 at 9:37
asmithasmith
11
New contributor
asmith is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked Jan 4 at 9:37
asmithasmith
11
New contributor
asmith is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked Jan 4 at 9:37
asmithasmith
11
asked Jan 4 at 9:37
asmithasmith
11
11
New contributor
asmith is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
asmith is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
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