# Mykhailo Pilipovich Krawtchouk

### Quick Info

Chovnitsy, (now Kivertsi) Ukraine

Kolyma, Siberia, USSR

**Mykhailo Pilipovich Krawtchouk**was a Soviet Ukrainian mathematician who worked on differential and integral equations.

### Biography

There are many different transliterations of Mikhail Krawtchouk's name, the forms Kravchuk and Krawtschuk also being used. The form used here "Krawtchouk" is the spelling used in the papers which he wrote in French. His father, Filipp Krawtschuk, graduated from Petrovsko-Razumovskaya Academy and became a land surveyor. His mother, born in Poland, was fluent in several European languages and taught Mikhail and his three siblings German, Polish, and French. But the main language the family spoke every day was their native Ukrainian.When Mikhail was nine years old, in 1901, the family moved to Lutsk, a city on the Styr River in northwest Ukraine. He attended the Lutsk Gymnasium and graduated with the Gold Medal. Krawtchouk then studied at the St Vladimir University in Kyiv and obtained his first degree in 1914. He published his first paper in 1914, namely

*On groups of commutative matrices*(Russian). The First World War broke out shortly after Krawtchouk graduated and, because of problems at Kyiv University it was evacuated to Saratov. Krawtchouk, however, chose to move to Moscow. A note dated 2 November 1916 contains his own report of his studies at Moscow University and this is given in [6]. The paper [6] also contains a report by Dmitry Aleksandrovich Grave, dated 21 February, giving an evaluation of Krawtchouk as a mathematician.

This, however, was a time of severe political problems with one disruption following another for Krawtchouk. In 1917 the Bolsheviks seized power in St Petersburg and fighting broke out in Moscow. Bolshevik power was soon firmly established and Krawtchouk returned to Kyiv. For two years Krawtchouk taught in a number of different institutions until the outbreak of the civil war. This was not an easy period for, in January 1918, an independent Ukrainian state was proclaimed with Kyiv as its capital. Red Army troops entered Kyiv in the following month. In Kyiv, Krawtchouk had met Esfira Iosifovna when they were both students [25]:-

They met for the first time during their student days, in Kiyanovskiy lane, where two sisters from Kyiv arranged dinners for students. They got married in 1918, taught together at the Savarian school.Mikhail Filippovich and Esfira Iosifovna Krawtchouk had two children, a son Zhenya and a daughter Natasha.

Later in 1918 an independent Ukraine was again declared in Kyiv but there followed a series of struggles between Ukrainian nationalist, White, and Red forces. In November 1919 Kyiv was briefly taken by the White armies before being occupied by the Red Army. There was still no peace in Kyiv for, in May 1920 the Poles captured Kyiv but were driven out in a counterattack. During this stormy period Krawtchouk was headmaster at a country school not far from Kyiv. It was in 1920 that Krawtchouk wrote his 96-page handwritten book

*Geometry for seven-year labour schools*. It was never printed or published and was only discovered in November 2005. The article [14] examines this book:-

Young talented mathematician already in the first years of his professional work didn't stand aside the issues of general education. He who has not only taught mathematics in secondary schools (I, II Ukrainian schools, electrical school, etc.), but also at the university, other institutions of Kyiv, being desperate about financial difficulties (which our entire intelligence experienced in that times) in the village Savarka (1920-1921), painfully feeling the lack of textbooks, undertook to conclude this book. He is working hard in the evenings and night hours, alternating between the research on higher mathematics (the theory of variable matrix transformations of quadratic forms, the theory of correlation, interpolation theory of functions of a real variable, etc.) and the best ways to apply elementary mathematics for students in grades 5-7.After working on his doctoral thesis advised by Dmitry Aleksandrovich Grave, he was awarded a doctorate for a dissertation

*On Quadratic Forms and Linear Transforms*in 1924. In the same year he attended the International Mathematical Congress in Toronto, giving the lecture "Note sur l'interpolation généralisée" Ⓣ, and made contacts with many mathematicians. Four years later he attended the International Mathematical Congress in Bologna where he gave the lecture "Sur l'integration approchée des équations différentielles linéaires"

Krawtchouk's contacts with other mathematicians were extremely valuable, particularly those with Jacques Hadamard, David Hilbert, Richard Courant and Francesco Tricomi. In 1929 Krawtchouk published his most famous work,

*Sur une généralisation des polynômes d'Hermite*Ⓣ. In this paper he introduced a new system of orthogonal polynomials now known as the Krawtchouk polynomials, which are polynomials associated with the binomial distribution. Let us note that, in March 2021, MathSciNet lists 112 papers with 'Krawtchouk' in the title, 88 of these having 'Krawtchouk polynomials' in the title. An additional 26 papers have 'Kravchuk' in their title. For more information about Krawtchouk polynomials, see [19] or [20]; this paper has the following review:-

Orthogonal Krawtchouk [Kravchuk] polynomials present a system of polynomials that are orthogonal with respect to the weight function which is the density of the binomial probability distribution. These polynomials belong to the class of orthogonal polynomials with discrete variables and were introduced by the Ukrainian mathematician M Krawtchouk in 1929. The content of original works by Krawtchouk devoted to this topic is described in detail here. A survey of the further developments and applications of Krawtchouk polynomials is provided.Although most references today are to Krawtchouk polynomials, his mathematical work was very wide and, despite his early death, he was the author of around 180 articles on mathematics. He wrote papers on differential and integral equations, studying both their theory and applications. The paper [14] by V O Movchan has the following summary:-

We analyse M P Kravchuk's work on generalising methods for the integration of ordinary differential equations, in particular the least squares method and the Ritz method, and on developing new methods for the approximate integration of differential, integral, and integro-differential equations. We prove that Kravchuk's scientific interest in this area was greatly stimulated by the studies of Walther Ritz and Nikolai Mitrofanovich Krylov.Other areas he wrote on included algebra (where among other topics he studied the theory of permutation matrices), geometry, mathematical and numerical analysis, probability theory and mathematical statistics. Eugene Seneta writes about his contributions to statistics in [23]:-

His work in mathematical statistics relates to the theory of correlation and regression, the bivariate normal, the method of moments in statistics, and orthogonal polynomial systems. Some of this was influenced by the work of the English Biometric School headed by Karl Pearson, as was the early work of Slutsky, also in Kyiv in the early years of his career. A theme running through much of Kravchuk's work, and not only in mathematical statistics, is the method of moments. His statistical direction was initially driven by the work of Chebyshev on interpolation and the method of moments, which in turn derived partly from Chebyshev's contact with the works of Bienaymé. In the Zapysky version of the 1929 paper, all the substantial papers of Chebyshev on interpolation are mentioned. Kravchuk, however, uses a more modern matrix approach in contrast to Chebyshev's continued fractions. Chebyshev's work was dominant in probability and statistics in the old czarist Russian empire, and this influence of the St Petersburg School is natural on a strong mathematician such as Kravchuk.He was also interested in the philosophy of mathematics, the history of mathematics and mathematical education. Krawtchouk edited the first three-volume dictionary of Ukrainian mathematical terminology [25]:-

Mikhail Krawtchouk was the first in the Ukraine to develop a project concerning algebraic and geometric terminology. Under his leadership, in the 1920s, employees of the Institute of the Ukrainian Scientific Language compiled a three-volume mathematical dictionary. And the students of Mikhail Filippovich, even after his death, will remember not only the beauty of the ideas that were presented in Krawtchouk's lectures, but also the beauty of his Ukrainian speech.In 1929 Krawtchouk was elected a full member of the Ukrainian Academy of Sciences. He taught at the Kyiv Polytechnic Institute (now the National Technical University of Ukraine) where he became a professor and was head of the Mathematical Chair. Surprisingly, despite continuing to publish deep research papers and having a high teaching and administrative load, he found time to publish articles on teaching mathematics in schools. For example he published the articles (all in Ukrainian and some with co-authors)

*Elements of the theory of determinants*(1933),

*A new method of teaching logarithms in secondary schools*(1936), and

*Approximate calculations in secondary schools*(1936). This, however, was a time of new political problems.

At this time political life in the USSR revolved round the exposure and suppression of alleged plotters against the regime. The country was subjected to an vigorous campaign against so-called enemies of the people. There was a series of public trials and in a massive terror campaign against the population as a whole. The worst phase of the terror took place during the period 1937-38 when at least five million people were sent to camps mostly in the Arctic. In 1937 Krawtchouk was accused of being a Polish spy and also a bourgeois nationalist [25]:-

The accusation was even that he had knowledge of foreign languages and corresponded with scholars of capitalist countries.He was arrested, tried and sentenced to twenty years in prison and five years in exile. As part of the punishment imposed on him, Krawtchouk was stripped of his membership of the Ukrainian Academy of Sciences and his exile was to be spent at the Kolyma camp in northeastern Siberia, one of the two most notorious of the Corrective Labour Camps. He was subjected to cruel torture and wrote to the chairman of the Supreme Court of the USSR (see [7]):-

I was stunned by these wild accusations, physically beaten by night interrogations, in particular, complete deprivation of sleep for eleven days, exacerbation of heart disease, measures of direct physical impact, morally affected by screams, moans of torture in neighbouring rooms ...For an English translation of [7] see THIS LINK.

His GULAG sentence is also the subject of [25]; for an English translation see THIS LINK.

He was finally broken by threats against his family: if he refused to accept that he had committed the crimes he was accused of, his family would be arrested and destroyed. He died at the age of 49 in Kolyma, one of the Labour Camps set up by the Gulag. As was typical of such events, Krawtchouk was written out of history at this time as if he had never existed. In 1956, however, he was rehabilitated but, although his work could now be spoken about, there would still be no mention of how he had met his death or where he died. For example, [30] written in 1968, gives an accurate account of his work, lists 161 of his publications, but concerning his death it simply states:-

The eminent scientist died on March 9, 1942, at the height of his creative powers and potential.In March 1992, fifty years after his death, he was restored as a member of the Ukrainian Academy of Sciences. In September 1992 the First International M Krawtchouk Conference was held in the Ukraine. Since then these conferences have been a yearly event. There was a special number of the

*Ukrainian Mathematical Journal*

**44**(7) 1992 dedicated to Krawtchouk; references [4], [5], [11], [12], [17], [18], [19], [20] all relate to this special issue [23]:-

Plaques in his memory have now been erected in Lutsk and in Kyiv, and a small museum established in part of the elementary school in Chovnytsia.Let us end this biography by quoting from Eugene Seneta about work of Krawtchouk which has hardly been noticed [23]:-

A little-noticed feature of Kravchuk's work is the expansion of a binomial probability value in terms of Krawtchouk polynomials orthonormal with respect to another binomial distribution. The germ of the idea is the analogous expansion of densities in terms of Hermite polynomials in Chebyshev's last probability paper (1887); and Kravchuk's expansion also produces the Charlier expansion of a binomial probability in the limit. More importantly, he gives a bound on the error of partial expansion, a very Chebyshevian idea. More importantly still, in his 1931 paper, he produces a polynomial system orthonormal with respect to the hypergeometric distribution. Subsequently this system, overlooked at the time outside Ukraine, even though it was described by Smohorshewsky in the very same French journal (C.R. Acad. Sci., Paris, 200 (1935) 801-803) has come to be credited to later work of Wolfgang Hahn. Inasmuch as the binomial distribution can be considered as a limit of the hypergeometric, this system generalises the Krawtchouk polynomials.

### References (show)

- Yu M Berezansskii, V S Korolyuk, I O Lukovskii, et al., On the 125th anniversary of the birth of the famous Ukranian mathematicial Mikhailo Pilipovich Kravchuk (27.09.1892-09.03.1942) (Ukrainian),
*Ukrain. Mat. Zh.***69**(9) (2017), 1265-1269. - V A Dobrovol'skii, Distinguished Ukrainian mathematician Mikhail Filippovich Kravchuk (on his 75th birthday),
*Uspekhi Mat. Nauk***23**(1) 1968, 236-239. - E M Dobrovolska, Continued fractions in the works of M P Kravchuk (Ukrainian) The Institute of Mathematics. Outlines of its development (Ukrainian),
*Trudi Inst. Mat. Nats. Akad. Nauk Ukraini*,**17**(Natsional. Akad. Nauk Ukraini, Inst. Mat., Kiev, 1997), 156-164. - M L Gorbachuk, Mikhail Pilipovich Kravchuk (on the occasion of the centenary of his birth) (Ukrainian),
*Ukrain. Mat. Zh.***44**(7) (1992), 859. - M L Gorbachuk, Mikhail Pilipovich Kravchuk (on the occasion of the centenary of his birth),
*Ukrainian Math. J.***44**(7) (1992), 771. - D A Grave and L V Kovalchuk, An account of professorial stipendiary Mikhail Kravchuk's work from November 1915 to November 1916 (Russian),
*Ocherki Istor. Estestvoznan. Tekhn. No.***37**(1989), 86-93. - S Gupalo, The fate of Mikhail Krawtchouk (Russian).

http://www.geocities.ws/orthpol/mirror.html - Ya F Kayuk, The role of the investigations of M V Ostragradskii and M P Kravchuk in the development of direct methods for solving problems in mathematical physics (Ukrainian), Ukrainian Mathematics Congress - 2001 (Ukrainian) (Natsional. Akad. Nauk Ukraini, Inst. Mat., Kiev, 2002), 48-63.
- M F Kravchuk, On the work of the Institute of Mathematics of the Academy of Sciences of the Ukrainian SSR,
*Uspekhi Mat. Nauk***3**(1937), 249-251. - M I Kratko (ed.),
*The Golgotha of Academician Kravchuk. A collection of documents*(Ukrainian) (Volinskii Institute of Post-Secondary Pedagogical Education, Lutsk, 2011). - T F Luchka and A Yu Luchka, Development of direct methods of mathematical physics in papers by M P Kravchuk (Ukrainian),
*Ukrain. Mat. Zh.***44**(7) (1992), 931-939. - T F Luchka and A Yu Luchka, Development of direct methods of mathematical physics in papers by M P Kravchuk,
*Ukrainian Math. J.***44**(7) (1992), 841-849. - V O Movchan, Mathematical statistics and probability theory in the works of M.P. Kravchuk (Ukrainian),
*Narysy Istor. Pryrodoznav. i Tekh.***17**(1972), 8-15. - V O Movchan, M P Kravchuk's work in the theory of differential and integral equations (Ukrainian),
*Narisi Istor. Prirodoznav. i Tekhn. No.***28**(1982), 3-12; 105. - O S Parasyuk and N O Virchenko, A short piece about the scientific heritage of M Krawtchouk,
*Ukrainian Mathematical Journal***44**(1992), 772-791. - O S Parasyuk and N O Virchenko, Brief survey of the mathematical legacy of Academician M Kravchuk,
*Fract. Calc. Appl. Anal.***12**(2) (2009), 115-134. - O S Parasyuk and N O Virchenko, A short piece about the scientific heritage of M Kravchuk (Ukrainian),
*Ukrain. Mat. Zh.***44**(7) (1992), 860-879. - O S Parasyuk and N O Virchenko, A short piece about the scientific heritage of M Kravchuk,
*Ukrainian Math. J.***44**(7) (1992), 772-791. - G I Prizva, Orthogonal Kravchuk polynomials (Ukrainian),
*Ukrain. Mat. Zh.***44**(7) (1992), 880-888. - G I Prizva, Orthogonal Kravchuk polynomials,
*Ukrainian Math. J.***44**(7) (1992), 792-800. - E Seneta, Krawtchouk polynomials and Australian statisticians,
*Institute of Mathematical Statistics Bulletin***22**(4) (1993), 421-423. - E Seneta, M Krawtchouk (1892-1942): Professor of mathematical statistics,
*Theory Stoch. Process***3**(1997), 388-392. - E Seneta, Mikhailo Pylypovych Kravchuk (or Krawtchouk). b. 27 September 1892 (o.s.) - d. 9 March 1942

https://encyclopediaofmath.org/images/e/e1/MikhailoPylypovychKRAVCHUKOrKRAWTCHOUK.pdf - E Sverstyuk, Scientist with the face of Christ (Ukrainian).

http://www.geocities.ws/orthpol/nasha_vira.html - O Unguryaw, Krawtchouk in the GULAG (Russian).

http://www.geocities.ws/orthpol/facty.html - V M Urbanskii,
*Mikhail Filippovich Kravchuk,*1892-1942? (Russian) (Nauka, Moscow, 2007). - N Virchenko, Mikhail Krawtchouk: Short Biography.

http://www.geocities.ws/orthpol/biogr.html - N Virchenko, M P Kravchuk's unknown handwritten textbook, National Technical University of Ukraine.

https://kpi.ua/en/618-3 - N A Virchenko, V A Dobrovolskii, Yu A Mitropolskii and A S Smogorzhevskii, Seventy-Fifth anniversary of the birth of Mikhail Filippovich Kravchuk (Ukrainian),
*Ukrainskii Matematicheskii Zhurnal***20**(1) (1968), 85-91. - N A Virchenko, V A Dobrovolskii, Yu A Mitropolskii and A S Smogorzhevskii, Seventy-Fifth anniversary of the birth of Mikhail Filippovich Kravchuk,
*Ukrainian Mathematical Journal***20**(1-2) (1968), 78-84. - V I Zelenkov and V A Savva, VI International Krawtchouk Conference: Kyiv, Ukraine, May 14-17, 1997, Orthogonal Polynomials and Special Functions Net.

http://www.math.yorku.ca/siamopsf/reports/ Kyiv.html

### Additional Resources (show)

Other pages about Mikhail Krawtchouk:

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Written by J J O'Connor and E F Robertson

Last Update March 2021

Last Update March 2021