Return to Answer

An analogous formula does hold, although the corresponding functions are not hypergeometric if $p$ is irrational.

For given $p\in\mathbb{R}$, $p>1$, consider the power series $$h(z)=\sum_{k=0}^{\infty} \frac{(-1)^k}{pk-k+1}\binom{pk}{k}\, z^k$$ with radius of convergence $R=(p-1)^{p-1}/p^p.$

Then, for $0\le y\le R^{1/(p-1)}$, the function $g(y):=yh(y^{p-1})$ is the inverse function of $f(x):=x+x^p$. $$*$$ [edit] There is also an analogous inversion formula for three or more terms, to invert e.g. $f(x)=x+ax^p+bx^q$ with real exponents $p>1$ and $q>1$. If $H$ is the analytic function $$H(u,v)=\sum_{i\ge0,j\ge0}\frac{(-1)^{i+j}}{ (p-1)i+ (q-1)j+1} {pi+qj \choose i,\, j}u^iv^j,$$ then $g(y):=yH(ay^{p-1},by^{q-1})$ is the local inverse of $f$ at $0$ (the multinomial coefficient in the double series is ${pi+qj \choose i,\, j}:=\frac{(pi+qj)(pi+qj-1)\dots(pi+qj-i-j+1)}{i!j!}$ .)

An analogous formula does hold, although the corresponding functions are not hypergeometric if $p$ is irrational.

For given $p\in\mathbb{R}$, $p>1$, consider the power series $$h(z)=\sum_{k=0}^{\infty} \frac{(-1)^k}{pk-k+1}\binom{pk}{k}\, z^k$$ with radius of convergence $R=(p-1)^{p-1}/p^p.$

Then, for $0\le y\le R^{1/(p-1)}$, the function $g(y):=yh(y^{p-1})$ is the inverse function of $f(x):=x+x^p$. $$*$$ [edit] There is also an analogous inversion formula for three or more terms, to invert e.g. $f(x)=x+ax^p+bx^q$ with real exponents $p>1$ and $q>1$. If $H=H_{p,q}$ is the analytic function $$H(u,v)=\sum_{i\ge0,j\ge0}\frac{(-1)^{i+j}}{ (p-1)i+ (q-1)j+1} {pi+qj \choose i,\, j}u^iv^j,$$ then $g(y):=yH(ay^{p-1},by^{q-1})$ is the local inverse of $f$ at $0$ (the multinomial coefficient in the double series is ${pi+qj \choose i,\, j}:=\frac{(pi+qj)(pi+qj-1)\dots(pi+qj-i-j+1)}{i!j!}$ .)

An analogous formula does hold, although the corresponding functions are not hypergeometric if $p$ is irrational.

For given $p\in\mathbb{R}$, $p>1$, consider the power series $$h(z)=\sum_{k=0}^{\infty} \frac{(-1)^k}{pk-k+1}\binom{pk}{k}\, z^k$$ with radius of convergence $R=(p-1)^{p-1}/p^p.$

Then, for $0\le y\le R^{1/(p-1)}$, the function $g(y):=yh(y^{p-1})$ is the inverse function of $f(x):=x+x^p$. $$*$$ There is also an analogous inversion formula for three or more terms, to invert e.g. $f(x)=x+ax^p+bx^q$ with real exponents $p>1$ and $q>1$. If $H$ is the analytic function $$H(u,v)=\sum_{i\ge0,j\ge0}\frac{(-1)^{i+j}}{ (p-1)i+ (q-1)j+1} {pi+qj \choose i,\, j}u^iv^j,$$ then $g(y):=yH(ay^{p-1},by^{q-1})$ is the local inverse of $f$ at $0$ (the multinomial in the double series is ${pi+qj \choose i,\, j}=\frac{(pi+qj)(pi+qj-1)\dots(pi+qj-i-j+1)}{i!j!}$ .)

An analogous formula does hold, although the corresponding functions are not hypergeometric if $p$ is irrational.

For given $p\in\mathbb{R}$, $p>1$, consider the power series $$h(z)=\sum_{k=0}^{\infty} \frac{(-1)^k}{pk-k+1}\binom{pk}{k}\, z^k$$ with radius of convergence $R=(p-1)^{p-1}/p^p.$

Then, for $0\le y\le R^{1/(p-1)}$, the function $g(y):=yh(y^{p-1})$ is the inverse function of $f(x):=x+x^p$. $$*$$ [edit] There is also an analogous inversion formula for three or more terms, to invert e.g. $f(x)=x+ax^p+bx^q$ with real exponents $p>1$ and $q>1$. If $H$ is the analytic function $$H(u,v)=\sum_{i\ge0,j\ge0}\frac{(-1)^{i+j}}{ (p-1)i+ (q-1)j+1} {pi+qj \choose i,\, j}u^iv^j,$$ then $g(y):=yH(ay^{p-1},by^{q-1})$ is the local inverse of $f$ at $0$ (the multinomial coefficient in the double series is ${pi+qj \choose i,\, j}:=\frac{(pi+qj)(pi+qj-1)\dots(pi+qj-i-j+1)}{i!j!}$ .)

An analogous formula does hold, although the corresponding functions are not hypergeometric if $p$ is irrational.

For given $p\in\mathbb{R}$, $p>1$, consider the power series $$h(z)=\sum_{k=0}^{\infty} \frac{(-1)^k}{pk-k+1}\binom{pk}{k}\, z^k$$ with radius of convergence $R=(p-1)^{p-1}/p^p.$

Then, for $0\le y\le R^{1/(p-1)}$, the function $g(y):=yh(y^{p-1})$ is the inverse function of $f(x):=x+x^p$. $$*$$ There is also an analogous inversion formula for three or more terms, to invert e.g. $f(x)=x+ax^p+bx^q$ with real exponents $p>1$ and $q>1$. If $H$ is the analytic function $$H(u,v)=\sum_{i\ge0,j\ge0}\frac{(-1)^{i+j}}{ (p-1)i+ (q-1)j+1} {pi+qj \choose i,\, j}u^iv^j,$$ then $g(y):=yH(ay^{p-1},by^{q-1})$ is the local inverse of $f$ at $0$.

An analogous formula does hold, although the corresponding functions are not hypergeometric if $p$ is irrational.

For given $p\in\mathbb{R}$, $p>1$, consider the power series $$h(z)=\sum_{k=0}^{\infty} \frac{(-1)^k}{pk-k+1}\binom{pk}{k}\, z^k$$ with radius of convergence $R=(p-1)^{p-1}/p^p.$

Then, for $0\le y\le R^{1/(p-1)}$, the function $g(y):=yh(y^{p-1})$ is the inverse function of $f(x):=x+x^p$. $$*$$ There is also an analogous inversion formula for three or more terms, to invert e.g. $f(x)=x+ax^p+bx^q$ with real exponents $p>1$ and $q>1$. If $H$ is the analytic function $$H(u,v)=\sum_{i\ge0,j\ge0}\frac{(-1)^{i+j}}{ (p-1)i+ (q-1)j+1} {pi+qj \choose i,\, j}u^iv^j,$$ then $g(y):=yH(ay^{p-1},by^{q-1})$ is the local inverse of $f$ at $0$ (the multinomial in the double series is ${pi+qj \choose i,\, j}=\frac{(pi+qj)(pi+qj-1)\dots(pi+qj-i-j+1)}{i!j!}$ .)