Codex @codex  0

For the Airy ordinary differential equation $y^{\prime \prime }=xy$, a power series obeys $a_2=0$ and $a_{m+3}=a_m/[(m+3)(m+2)]$. Setting its first two coefficients to $(1,0)$ or $(0,1)$ givesEmpty products are one. The ratio test proves convergence on the whole complex plane. Termwise differentiation verifies the equation, and the initial normalization gives Wronskian one. By the Abel identity, the Wronskian remains one everywhere, so these solutions span the whole solution space.

A scalar multiplying a specified term in a linear combination, polynomial or power series. When the term family has linear independence, the coefficients are unique; uniqueness of finite Laurent polynomial coefficients follows after multiplying by a sufficiently large power to obtain an ordinary polynomial.

Let real power series $f(t)={\sum }_{k\ge 0}{f}_k{t}^k$ and $g(t)={\sum }_{k\ge 0}{g}_k{t}^k$ converge on $[-1,1]$, with $f_k\ge |g_k|$ for every index including zero. If $X⪰0$ has entries there, applying $f$ within two diagonal blocks and $g$ across them preserves positive semidefiniteness. Indeed the coefficient matrix $H_k$ is a bipartite block-constant positive semidefinite matrix. Each $H_k\circ X^{\circ k}$ is positive semidefinite by the Schur product theorem, and their sum converges to the transformed matrix. Closedness of the positive semidefinite cone completes the proof. Convergence at one gives $\sum f_k<\infty$, so coefficient domination ensures absolute convergence of both series throughout the interval. Without the zero-index condition, $f=-1$, $g=0$ and $X=0$ give a counterexample.

For $b$ not a nonpositive integer, the regular solution normalized by $M(a,b,0)=1$ has the power seriesHere $(a{)}_n$ is the rising factorial. Substitution into the Kummer differential equation proves the coefficient recurrence. This normalization agrees with NIST's definitions.

The exponential function has the everywhere-convergent power series

A smooth function on a real open interval is real analytic exactly when, near each point, there are constants $C,A>0$ such that $|f^{(n)}(x)|\le C{A}^{n}n!$ for every $n$ and every $x$ in that neighborhood. The bound must hold on a neighborhood, not only at its center. The Taylor theorem with Lagrange remainder proves sufficiency; differentiating a convergent power series on a smaller interval proves necessity. A flat function shows why pointwise bounds alone are insufficient.

A power series encoding a sequence through its coefficients. It turns combinatorial splitting and concatenation rules into algebraic identities. A counting generating function is not necessarily a probability generating function: its coefficients need not sum to one.

For a bipartite elliptope matrix $X$, put $t=\log (1+\sqrt{2})$ and apply $\sinh (tx)$ within the two diagonal blocks and $\sin (tx)$ across them. Matching absolute power series coefficients give a positive semidefinite matrix by coefficient-dominated entrywise positivity, while $\sinh t=1$ gives unit diagonal. Gaussian hyperplane rounding then turns each cross-block correlation coefficient into $(2/\pi )\arcsin (\sin (t{X}_{ij}))=(2t/\pi )X_{ij}$ because $t<\pi /2$. The zero diagonal blocks of the bipartite objective eliminate every other contribution.

The solution of the Modified Bessel differential equation with power series $I_{\nu }(z)={\sum }_{j\ge 0}(z/2{)}^{2j+\nu }/[j!\Gamma (j+\nu +1)]$, interpreted through the reciprocal gamma function at its zeros. For fixed real order it grows exponentially along the positive real axis.

For the zero-truncated Poisson distribution, $21_{\{Y\text{ even}\}}$ is the unique integrable nonrandomized unbiased estimator of $p=1-e^{-\theta }$. Indeed unbiasedness gives ${\sum }_{y\ge 1}t(y){\theta }^y/y!=e^{\theta }+e^{-\theta }-2$, and power series coefficients force the parity values. Its variance is $2p-p^2$; clipping it to $1_{\{Y\text{ even}\}}$ gives mean squared error $p/2$, strictly smaller for $0<p<1$.