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Frank W. J. Olver worked on special functions, asymptotic analysis of differential equations and reliable numerical methods. His research relates limiting descriptions of solutions to the computation of functions such as Bessel functions and hypergeometric functions. He authored the integer-order Bessel chapter in the original NIST handbook and helped develop the modern NIST Digital Library of Mathematical Functions.^[ref]

Homepage: dlmf.nist.gov/about/bio/FWJOlver

Leonard C. Maximon developed mathematical tools for quantum electrodynamics and contributed to the NIST Digital Library of Mathematical Functions. His authored chapter on angular-momentum symbols organizes the coefficients needed to combine and recouple quantum angular momenta. These coefficients connect alternative coupled bases, so their identities and numerical evaluation matter in spectroscopy and scattering calculations. He also coauthored the library's treatment of Bessel functions, linking reusable special functions with physical applications.^[ref]

Homepage: dlmf.nist.gov/about/bio/LCMaximon

Nico M. Temme studies the asymptotic analysis and numerical methods for evaluating of special functions. His work includes asymptotic expansions for integrals and computational methods for functions such as the error function. The NIST Digital Library of Mathematical Functions credits him with chapters on numerical methods and several special function families.^[ref]

Homepage: dlmf.nist.gov/about/bio/NMTemme

A Painlevé transcendent is a special function obtained as a solution of one of the six nonlinear second-order Painlevé equations. The family is organized by the differential equation, its parameters and its initial or asymptotic data. In general these solutions supply new transcendental functions, rather than combinations of classical special functions.^[ref]

The name of the family does not imply that every exceptional solution is a transcendental function. For the Painlevé II equation, $y=0$ at $\alpha =0$ and $y=-1/x$ at $\alpha =1$ are rational solutions. The Painlevé property concerns the absence of movable branch points, rather than the absence of movable poles; it describes how solutions behave under analytic continuation.

A credited contributor to Chapter 1 of the NIST Digital Library of Mathematical Functions, covering algebraic and analytic methods. His chapter contribution connects elementary identities with the methods used to study special functions.^[ref]

A special function is a named family of functions with a developed theory of identities, transformations, asymptotic behavior and numerical evaluation, often arising repeatedly as solutions of differential equations or integral representations. The boundary with elementary functions is conventional rather than a strict mathematical classification.

For example, the regular solution of $x^2{y}^{\prime \prime }+x{y}^{\prime }+x^{2}y=0$ normalized by $y(0)=1$ is the order-zero Bessel function $J_0$. Such a function packages a recurring radial-wave calculation into a reusable object. The Gamma function similarly extends factorial values through $\Gamma (n+1)=n!$ for nonnegative integers, allowing identities involving integer parameters to be studied continuously.