**Nikolay Sonin**'s father Yakov Sonin had trained in law and was working as a state official at the time of Nikolay's birth but later practiced as a lawyer. Nikolay Sonin attended Moscow University where he studied mathematics and physics in the Faculty of Physics and Mathematics from 1865 to 1869. In the middle of this period Nicolai Vasilievich Bugaev, after studying for a period of two and a half years with Kummer and Weierstrass in Berlin and Liouville in Paris, was appointed as a professor at Moscow University. Sonin published his first paper on differentiation of functions defined with complex exponents in 1869, the year he obtained his first degree. Sonin then undertook research under Bugaev's supervision. He obtained a Master's Degree with a thesis on the expansion of functions in infinite series submitted in 1871. He then taught at the University of Warsaw where he was appointed as a dozent in 1872. He continued working on his doctorate, essentially equivalent to the German habilitation, and after submitting a thesis on partial differential equations of the second order to the Moscow University he was awarded the degree in 1874.

In 1876 Sonin was appointed to a chair in the University of Warsaw. There he played several major roles in high positions in the University including two spells when he was elected to serve as dean of the Faculty of Physics and Mathematics. He also organised the Society of Natural Scientists and was elected to the Russian Academy of Sciences in 1891. Two years later he was made an academician in pure mathematics and moved to St Petersburg to take part in the academic and administrative work of the Academy. From 1894 Sonin also taught at the St Petersburg University for Women. From 1899 to 1901 he was superintendent of the St Petersburg Educational District and then from 1901 he was president of the Scientific Committee of the Ministry of National Education. At Klein's invitation he joined the International Committee on Mathematical Education (1909-1914).

Sonin worked on special functions, in particular studying functions such as Bessel functions, gamma functions and their generalizations, such as functions *F*(*x*) satisfying *F*(*x*+1) - *F*(*x*) = *f* (*x*). In particular he undertook research on cylindrical functions and also worked on the Euler-Maclaurin summation formula. Among other topics to which he made contributions are Bernoulli polynomials (named after Jacob Bernoulli), mechanical quadrature, approximate computation of definite integrals in which he continued Chebyshev's work on numerical integration, Chebyshev inequalities and multiple integrals. He has a sequence of polynomials named after him - the Sonin polynomials *T*^{n}_{m}(*x*) satisfy the differential equation

Youschkevitch writes in [1]:-xy" + (m+ 1 -x)y' +ny= 0.

Together with A A Markov, Sonin prepared a two volume edition of Chebyshev's works in French and Russian.Sonin made a substantial contribution to the theory of special functions; the unifying idea of his researches was to establish a few convenient definitions of initial notions and operations leading to broad and fruitful generalisations of the these functions. Especially important were his discoveries in the theory of cylindrical functions, which he enriched both with general principles and with many particular theorems and formulas that he introduced into the contemporary literature. He also worked on Bernouillian polynomials, and his works on the general theory of orthogonal polynomials were closely interwoven with his research on the approximate computation of definite integrals and on various integral inequalities; in the latter area he continued Chebyshev's research.

**Article by:** *J J O'Connor* and *E F Robertson*

**Click on this link to see a list of the Glossary entries for this page**