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In a bipartite graph, a self-avoiding walk ends in its starting class exactly when it has even length. If degrees are bounded by $\Delta$, deleting the last edge gives ${\sigma }_{2k+1}\le \Delta {\sigma }_{2k}$ and hence $Z_G^0(x)\le Z_G(x)\le (1+\Delta x)Z_G^0(x)$ for $x\ge 0$. Thus these generating functions have the same radius of convergence.

A power series is lacunary when only a sparse sequence of powers has nonzero coefficients. For ${\sum }_j{c}_j{z}^{m_j}$ with strictly increasing integer exponents, the radius of convergence satisfies $R^{-1}={\mathrm{limsup}}_j|c_j{|}^{1/m_j}$ by the Cauchy-Hadamard theorem. The exponent $m_j$, rather than the term index $j$, controls the coefficient root.

The radius of convergence of a power series is the extended real number $R\in [0,\infty ]$ characterized by absolute convergence for $|x|<R$ and divergence for $|x|>R$. Equivalently,For $R=0$, only the central point is guaranteed; for $R=\infty$, every real point is inside. At finite endpoints $x=\pm R$, convergence must be investigated separately and is not determined by $R$.

The termwise differentiation of a power series theorem states that inside its radius of convergence, the sum is differentiable and its derivative is the sum of the differentiated series. Accordingly $\boxed{f^{\prime }(x)=g(x)\text{ for }|x|<R}$. There is no blanket differentiability assertion at the convergence endpoints.

Since the common radius of convergence is infinite, the termwise differentiation of a power series theorem applies on the entire real line. For the even series, differentiation and the index change $m=n-1$ giveThe odd series similarly gives $g^{\prime }(x)={\sum }_{n=0}^{\infty }(-1{)}^n{x}^{2n}/(2n)!=f(x)$. Thus $\boxed{f^{\prime }=-g,g^{\prime }=f}$ everywhere. By the product rule,The mean value theorem implies that a differentiable function with zero derivative is constant on the real line. At zero, the series give $f(0)=1$ and $g(0)=0$. Consequently $\boxed{f(x{)}^2+g(x{)}^2=1\text{ for every }x\in \mathbb{R}}$. This derives the identity from the power series without assuming a trigonometric identity in advance.

The bipartite graph property means that a walk starting in $V_0$ ends there precisely when its length is even. Thus the even-length walk generating function on a bipartite graph is $Z_G^0(x)={\sum }_{k\ge 0}{\sigma }_{2k}{x}^{2k}$. Deleting the last edge of an odd-length self-avoiding walk leaves an even-length one; each such prefix has at most $\Delta$ possible last-edge choices. Hence ${\sigma }_{2k+1}\le \Delta {\sigma }_{2k}$, and for $x\ge 0$,These inequalities also hold with infinite values. Their finite positive prefactor shows that the two power series have the same radius of convergence $1/\mu$; alternatively apply the root formula to the even subsequence.