##
**Development of mathematical ideas of Mykhailo Kravchuk.
(Розвыток математычных идей Мыхайла Кравчука.)**
*(Ukrainian.
English summary)*
Zbl 1107.00012

Kyïv: N’yu-Jork (ISBN 0-88054-141-5). xlviii, 780 p. (2004).

From the introduction by Nina Opanasivna Virchenko: This book is dedicated to the presentation of the development and application of mathematical ideas of an outstanding twentieth century Ukrainian mathematician, Mykhailo Kravchuk (Krawtchouk). A full member of the Ukrainian Academy of Sciences (Ukrainian Academy of Sciences was known by different names during different periods of its history: Ukrainian Academy of Sciences (abbreviation in Ukrainian: UAN) 1918–21, All-Ukrainian Academy of Sciences (YUAN) 1921–36, Academy of Sciences of the Ukrainian SSR (AN URSR) 1936–1992, National Academy of Sciences of Ukraine (NANU) since 1992), Mykhailo Kravchuk made significant contributions to various branches of mathematics. He obtained in particular fundamental results in algebra, mathematical analysis, theory of differential and integral equations, probability and mathematical statistics. All of his multifaceted creative and public life was closely connected with the Ukrainian scientific and educational institutions.

Mykhailo Kravchuk was born on 27th of September 1892 in Chovnytsya, a village in the Volyn region. His father was a land surveyor. The boy received his primary education at home. In 1901 the family moved to the city of Lutsk. There, in 1910, Mykhailo completed his secondary education, graduating with highest honours. That same year he was admitted as a student to the department of mathematics and physics at St. Volodymyr University in Kyiv. After completing his undergraduate work in 1914, Kravchuk received an assistantship from the University to continue his studies in the department of mathematics and physics, in order to prepare for an academic career of research and teaching. During the years 1915–17, he completed his master’s degree and published several articles on linear algebra and Ukrainian mathematical terminology. The eventful year 1917 saw the demise of the Russian monarchy and its despotic regime. That same year saw the establishment of an Ukrainian state and the ensuing struggle for independence of Ukraine. Mykhailo Kravchuk did not stand aside from these historical events in the life of his nation. While actively pursuing his research, he also devoted much effort to help in the development of Ukrainian educational and scientific establishments. He participated in the work of the Ukrainian Scientific Society in Kyiv, the Mathematics-Physics Society of Kyiv University and taught at the newly founded 1st and 2nd Ukrainian gymnasia in Kyiv (first of their kind in the capital of Ukraine).

When the Ukrainian Academy of Sciences was founded in 1918, he became a research worker the Academic institutions and, in early 1920th, served on the Mathematical Terminology Committee of the Academy. During 1920-21, when instability and chaos were rampant after revolution and war of independence, Mykhailo Kravchuk left the city and found work as a principal and teacher at a secondary school in the village Savarka near Kyiv. In the following years, upon his return to Kyiv, Kravchuk was invited to lecture at various educational institutions in Kyiv University, polytechnic institute, architectural institute, veterenary-zootechnic, agricultural, aviation institute, and others). Besides his pedagogical activities, Kravchuk devoted himself wholeheartedly to research.

In 1924 he successfully defended his doctoral dissertation “On Quadratic Forms and Linear Transformations” before the Scientific Commission of the Ukrainian Academy of Sciences. In 1925 he was elected to Shevchenko Scientific Society (Lviv); in 1926 - 1927 he became a member of the Mathematical Societies of Germany, France, Palermo in Italy. In 1928 Kravchuk presented his research results at the International Mathematical Congress in Bologna, Italy. The following year, on June 29, 1929, Mykhailo Kravchuk was unanimously elected full member of the Ukrainian Academy of Sciences. Intense research activities resulted in the publication of his papers abroad. Mykhailo Kravchuk had contacts with mathematicians in France, Germany, Italy and other countries, including: J. Hadamard, R. Courant, N. Luzin, F. Tricomi and T. Levi-Civita. At the Ukrainian Academy of Sciences, M. Kravchuk held the position of scientific secretary of UAN. He also was dean of the faculty of professional education of Kyiv University and a member of the scientific board of the City Council of Kyiv. During 1934– 38 M. Kravchuk directed the Section of Mathematical Statistics at the Institute of Mathematics of the Ukrainian Academy of Sciences.

The 1930s were the years of great terror and massive repression in USSR and particularly in Ukraine. Soviet authorities were bent on destroying the nascent Ukrainian renaissance. Kravchuk’s dedication to the development of science in Ukraine, his influence and popularity among young scientists and University students did not go unnoticed by the authorities. Mykhailo Kravchuk was arrested on the 21st of February 1938 and charged with Ukrainian nationalism, counterrevolutionary activities and espionage. These were stereotype charges ascribed to many of those who were arrested during that time. On the 28th of September, M. Kravchuk was sentenced to 20 years imprisonment, followed by additional 5 years in which he was to be deprived of all civil rights. He was sent to a concentration camp in Kolyma, remote northeast of the Soviet Union, to serve sut his sentence. There, as a prisoner, Kravchuk spend three years at hard labour. With his health ruined by the extreme conditions of hunger, cold, physical and mental deprivation, M. Kravchuk died on March 9, 1942 and was buried in the Kolyma permafrost. On September 15, 1956 the Soviet authority “rehabilitated” M. Kravchuk, since no evidence could be found that he had committed a crime. But only on the 20th of March 1992, almost 100 years after his birth, Mykhailo Kravchuk was readmitted to membership in the National Academy of Sciences of Ukraine (NANU). The same year his name was entered in the International Calendar of Scientists by UNESCO. The First Kravchuk International Conference was held at Kyiv Polytechnic Institute in 1992. Since that time there were nine such conferences. The tenth Kravchuk conference is being organized to take place in Kyiv in May 2004. Lately two books of M. Kravchuk’s works were published in Kyiv.

The first, ”Popular Scientific Works,” appeared in 2000; the second, ”Selected Mathematical Works” 792 p. (2002; Zbl 1042.01017), was published jointly with the Ukrainian Academy of Arts and Sciences in the USA. On the 16-th of May 2002, the National Technical University of Ukraine (formerly known as Kyiv Polytechnic Institute) named an auditorium in his honour, and on the 20th of May 2003, the NTUU unveiled a statue of M. Kravchuk. M. Kravchuk authored more than 180 scientific works, including 10 monographs, in various branches of mathematics: algebra, number theory, theory of functions of real and complex variables, probability, statistics and history of mathematics.

The fundamental areas of his research were: investigation of the theory of commutative matrices, quadratic and bilinear forms, linear transformations, theory of algebraic and transcendental equations; number theory; investigation of certain problems in the theory of real functions and functions of complex variable; on interpolation methods, development of the least squares method applied to the solution of differential and integral equations; the creation and mathematical proof of the generalized method of moments and its application to the approximate solutions of ordinary linear differential equations, integral equations and equation of mathematical physics (American scientists an inventor John Atanasoff employed these results in the making of the first electronic digital computer); development of correlation theory, application of the method of moments in mathematical statistics; introduction and use of polynomials associated with the binomial distribution, now known in the mathematical literature as the Kravchuk polynomials.

The present volume contains selected articles of well-known mathematicians from the US, France, Germany, Great Britain, Netherlands, Australia, Portugal, India, Japan, Russia, Ukraine and other countries, who used Kravchuk’s mathematical results and ideas to further mathematical knowledge. They are from such diverse fields as algebra, number theory, mathematical physics and coding theory. It is noteworthy, that despite the fact that Mykhailo Kravchuk’s name and works were seldom mentioned after his incarceration by the Soviet regime, his scientific ideas did reach the broad mathematical world and gave rise to further development in the field. At the end of this book is a list of publications dedicated to the development of Kravchuk’s mathematical ideas. The list is selective. While it is impossible to list all such works, the bibliography is of sufficient magnitude to allow interested scientists to become acquainted with works by Kravchuk and to advance the ideas that are Mykhailo Kravchuk’s mathematical legacy.

Contents of the book:

Parasyuk, O.; Virchenko, N. [Brief survey of the mathematical legacy of academician M. Kravchuk, p. xv-xxxv]; [List of the works of M. Kravchuk. Compilers N. Virchenko, H. Syta, xxxvi-xlviii].

Chapter 1. Algebra. Physics.

Dunkl, Charles F.; Ramirez, Donald E. [Krawtchouk polynomials and the symmetrization of hypergroups. p. 3–18, see [SIAM J. Math. Anal. 5, 351-366 (1974; Zbl 0249.43006)]];

Stanton, Dennis [Some \(q\)-Krawtchouk polynomials on Chevalley groups, p. 19–56 [Am. J. Math. 102, 625–662 (1980; Zbl 0448.33019)]];

Stanton, Dennis [Three addition theorems for some \(q\)-Krawtchouk polynomials, p. 57–79 [Geom. Dedicata 10, 403–425 (1981; Zbl 0497.43006)]];

Koornwinder, Tom H. [Krawtchouk polynomials, a unification of two different group theoretic interpretations, p. 80–92 [SIAM J. Math. Anal. 13, 1011–1023 (1982; Zbl 0505.33015)]];

Schempp, Walter [Extensions of the Heisenberg group and coaxial coupling of transverse igenmodes, p. 93–103 [Rocky Mt. J. Math. 19, No. 1, 383–394 (1989; Zbl 0723.22009)]];

Koornwinder, Tom H. [Askey-Wilson polynomials as zonal spherical functions on the \(SU(2)\) quantum group, p. 104–122 [SIAM J. Math. Anal. 24, No. 3, 795–813 (1993; Zbl 0799.33015)]];

Feinsilver, Philip; Fitzgerald, Robert [The spectrum of symmetric Krawtchouk matrices, p. 123–141 [Linear Algebra Appl. 235, 121–139 (1996; Zbl 0845.15007)]];

Ericson, Thomas; Simonis, Juriaan; Tarnanen, Hannu; Zinoviev, Victor [\(F\)-partitions of cyclic groups, p. 142–148 [Appl. Algebra Eng. Commun. Comput. 8, No. 5, 387–393 (1997; Zbl 0893.20039)]];

Zhedanov, Alexei [\(9j\)-symbols of the oscillator algebra and Krawtchouk polynomials in two variables, p. 149–165 [J. Phys. A, Math. Gen. 30, No. 23, 8337–8353 (1997; Zbl 0952.33010)]];

Roy, B.; Roy, P. [Coherent states, even and odd coherent states in a finite-dimensional Hilbert space and their properties, p. 166–176 [J. Phys. A, Math. Gen. 31, No. 4, 1307–1317 (1998; Zbl 0907.47061)]];

Hakioğlu, Tuğrul; Wolf, Kurt Bernardo [The canonical Kravchuk basis for discrete quantum mechanics, p. 177–187 [J. Phys. A, Math. Gen. 33, No. 16, 3313–3323 (2000; Zbl 1052.81553)]];

Koelink, H. T. [\(q\)-Krawtchouk polynomials as spherical functions on the Hecke algebra of type \(B\), p. 188-212 [Trans. Am. Math. Soc. 352, No. 10, 4789–4813 (2000; Zbl 0957.33014)]];

Klimyk, A. U. [Krawtchouk polynomials, \(q\)-Krawtchouk polynomials and representation theory. p. 213–229];

Atakishiyev, M. N.; Groza, V. A. [Operator approach to quantum and affine \(q\)-Krawtchouk polynomials, p. 230–241];

Savva, V. A.; Zelenkov, V. I.; Khlus, O. V. [Quantum Krawtchuouk oscillators, their dynamics under laser radiation, p. 242–258].

Chapter 2. Special functions. Mathematical physics.

Rahman, Mizan [An elementary proof of Dunkl’s addition theorem for Krawtchouk polynomials, p. 261–268 [SIAM J. Math. Anal. 10, 438–445 (1979; Zbl 0414.33007)]];

Morrison, J. A. [Weighted averages of Radon transforms on \({\mathbb Z}_ 2^ k\), p. 269–278 [SIAM J. Algebraic Discrete Methods 7, 404–413 (1986; Zbl 0587.42007)]];

Zeng, Jiang [Calcul Saalschützien des partitions et des dérangements colorés, p. 279–286 [Saalschütz calculus of partitions and coloured derangements, SIAM J. Discrete Math. 3, No. 1, 149–156 (1990; Zbl 0703.05003)]];

Zeng, Jiang [Linearization of products of polynomials of Meixner, Krawtchouk, and Charlier, p. 287–306 [SIAM J. Math. Anal. 21, No. 5, 1349–1368 (1990; Zbl 0724.33006)]];

Dette, Holger [New bounds for Hahn and Krawtchouk polynomials, p. 307-319 [SIAM J. Math. Anal. 26, No. 6, 1647–1659 (1995; Zbl 0842.33005)]];

Abdelkarim, F.; Maroni, P. [The \(D_ \omega\)-classical orthogonal polynomials, p. 320-347 [Result. Math. 32, No. 1-2, 1–28 (1997; Zbl 0889.42020)]];

Sookoo, Norris [Generalized Krawtchouk polynomials: New properties, p. 348-354 [Arch. Math., Brno 36, No. 1, 9–16 (2000; Zbl 1055.33011)]];

Yakubovich, S. [Convolution Hilbert spaces associated with the Kontorovich - Lebedev transformation. p. 355–363];

Dunkl, C. [Singular polynomials for the symmetric group and Krawtchouk polynomials. p. 363–373];

Chapter 3. Probability theory and mathematical statistics.

Hoare, M. R.; Rahman, Mizan [Cumulative Bernoulli trials and Krawtchouk processes, p. 389–415 [Stochastic Processes Appl. 16, 113–139 (1984; Zbl 0534.60060)]];

Feinsilver, Philip; Schott, René [Krawtchouk polynomials and finite probability theory, p. 416–423 [Algebraic structures and operator calculus. Vol. II: Special functions and computer science. Mathematics and its Applications (Dordrecht) 292. Dordrecht: Kluwer Academic Publishers. ix, 148 p. (1994; Zbl 1128.33300)]];

Seneta, Eugene [M. Krawtchouk (1892–1942) – Professor of mathematical statistics, p. 424-429 [Theory Stoch. Process. 3(19), No. 3-4, 388–392 (1997; Zbl 0932.01037)]];

Seneta, Eugene [Characterization of Markov chains by orthogonal polynomial systems. p. 430–433];

Korolyuk, V. S.; Makarov, V. L.; Strok, V. V. [Development of ideas of M. Krawtchouk in the theory of orthogonal polynomials. p. 434-449];

Chapter 4. Coding theory. Theory of signals. Number theory.

Odlyzko, A. M.; Sloane, N. J. A. [A theta-function identity for nonlattice packings, p. 453–457 [Stud. Sci. Math. Hung. 15, 461–465 (1980; Zbl 0471.52007)]];

Bromba, M.; Ziegler, H. [Explicit formula for filter function of maximally flat nonrecursive digital filters, p. 458–461 [Electron. Lett. 16, No. 24, 905–906 (1980)]];

Bromba, M.; Ziegler, H. [Further results on maximally flat nonrecursive digital filters, p. 462–466 [Electron. Lett. 18, No. 23, 1014–1015 (1982)]];

Camion, P.; Courteau, B.; Fournier, G.; Kanetkar, S. V. [Weight distribution of translates of linear codes and generalized Pless identities, p. 467-489 [J. Inf. Optim. Sci. 8, 1–23 (1987; Zbl 0633.94018)]];

Tolhuizen, L.; van Lint, J. H. [On the minimum distance of combinatorial codes, p. 490–493 [IEEE Trans. Inf. Theory 36, No. 4, 922–923 (1990; Zbl 0707.94014)]];

Solé, Patrick [An inversion formula for Krawtchouk polynomials with applications to coding theory, p. 494-500 [J. Inf. Optimization Sci. 11, No. 2, 207–213 (1990; Zbl 0708.05063)]];

Solé, Patrick [Packing radius, covering radius, and dual distance, p. 501–515 [IEEE Trans. Inf. Theory 41, No. 1, 268–272 (1995; Zbl 0830.94022)]];

Levenshtein, Vladimir I. [Krawtchouk polynomials and universal bounds for codes and designs in Hamming spaces, p. 516–570 [IEEE Trans. Inf. Theory 41, No. 5, 1303–1321 (1995; Zbl 0836.94025)]];

Suen, Chung-yi; Chen, Hegang; Wu, C. F. J. [Some identities on \(q^{n-m}\) designs with application to minimum aberration designs, p. 571–583 [Ann. Stat. 25, No. 3, 1176–1188 (1997; Zbl 0898.62095)]];

Habsieger, Laurent; Stanton, Dennis [More zeros of Krawtchouk polynomials, p. 584–593 [Graphs Comb. 9, No. 2, 163–172 (1993; Zbl 0805.33008)]];

Krasikov, Ilia; Litsyn, Simon [On integral zeros of Krawtchouk polynomials, p. 594–622 [J. Comb. Theory, Ser. A 74, No. 1, 71–99 (1996; Zbl 0853.33008)]];

Stroeker, Roelof J.; de Weger, Benjamin M. M. [On integral zeroes of binary Krawtchouk polynomials, p. 623–633 [Nieuw Arch. Wiskd., IV. Ser. 17, No. 2, 175–186 (1999; Zbl 1069.33010)]];

Krasikov, Ilia [Bounds for the Christoffel-Darboux kernel of the binary Krawtchouk polynomials, p. 634–639 [Barg, Alexander (ed.) et al., Codes and association schemes. DIMACS workshop, DIMACS Center, Princeton, NJ, USA, November 9-12, 1999. Providence, RI: AMS, American Mathematical Society. DIMACS, Ser. Discrete Math. Theor. Comput. Sci. 56, 193–198 (2001; Zbl 0972.33006)]];

Krasikov, Ilia; Litsyn, Simon [Survey of binary Krawtchouk polynomials, p. 640–654 [Barg, Alexander (ed.) et al., Codes and association schemes. DIMACS workshop, DIMACS Center, Princeton, NJ, USA, November 9-12, 1999. Providence, RI: AMS, American Mathematical Society. DIMACS, Ser. Discrete Math. Theor. Comput. Sci. 56, 199–211 (2001; Zbl 0981.33003)]].

Chapter 5. Method of moments.

Katchanovski, I. [Mykhailo Krawtchouk and a puzzle in the invention and patenting of the electronic computer in US. p. 655–687];

Luchka, A. Yu.; Luchka, T. F. [Mykhailo Krawtchouk and variational and projection methods. p. 688-709];

Chyp, M. M. [The moment problem and integral representation of analytic functions. p. 710–732];

Kayuk, Ya. F. [Analysis of Mykhailo Krawtchouk’s method of moments on the basis of variation correlations of the mechanics of a continuous medium. p. 733–747];

Katchanovski, I. [List of the works that develop M. Kravchuk’s mathematical ideas. p. 748–775].

Mykhailo Kravchuk was born on 27th of September 1892 in Chovnytsya, a village in the Volyn region. His father was a land surveyor. The boy received his primary education at home. In 1901 the family moved to the city of Lutsk. There, in 1910, Mykhailo completed his secondary education, graduating with highest honours. That same year he was admitted as a student to the department of mathematics and physics at St. Volodymyr University in Kyiv. After completing his undergraduate work in 1914, Kravchuk received an assistantship from the University to continue his studies in the department of mathematics and physics, in order to prepare for an academic career of research and teaching. During the years 1915–17, he completed his master’s degree and published several articles on linear algebra and Ukrainian mathematical terminology. The eventful year 1917 saw the demise of the Russian monarchy and its despotic regime. That same year saw the establishment of an Ukrainian state and the ensuing struggle for independence of Ukraine. Mykhailo Kravchuk did not stand aside from these historical events in the life of his nation. While actively pursuing his research, he also devoted much effort to help in the development of Ukrainian educational and scientific establishments. He participated in the work of the Ukrainian Scientific Society in Kyiv, the Mathematics-Physics Society of Kyiv University and taught at the newly founded 1st and 2nd Ukrainian gymnasia in Kyiv (first of their kind in the capital of Ukraine).

When the Ukrainian Academy of Sciences was founded in 1918, he became a research worker the Academic institutions and, in early 1920th, served on the Mathematical Terminology Committee of the Academy. During 1920-21, when instability and chaos were rampant after revolution and war of independence, Mykhailo Kravchuk left the city and found work as a principal and teacher at a secondary school in the village Savarka near Kyiv. In the following years, upon his return to Kyiv, Kravchuk was invited to lecture at various educational institutions in Kyiv University, polytechnic institute, architectural institute, veterenary-zootechnic, agricultural, aviation institute, and others). Besides his pedagogical activities, Kravchuk devoted himself wholeheartedly to research.

In 1924 he successfully defended his doctoral dissertation “On Quadratic Forms and Linear Transformations” before the Scientific Commission of the Ukrainian Academy of Sciences. In 1925 he was elected to Shevchenko Scientific Society (Lviv); in 1926 - 1927 he became a member of the Mathematical Societies of Germany, France, Palermo in Italy. In 1928 Kravchuk presented his research results at the International Mathematical Congress in Bologna, Italy. The following year, on June 29, 1929, Mykhailo Kravchuk was unanimously elected full member of the Ukrainian Academy of Sciences. Intense research activities resulted in the publication of his papers abroad. Mykhailo Kravchuk had contacts with mathematicians in France, Germany, Italy and other countries, including: J. Hadamard, R. Courant, N. Luzin, F. Tricomi and T. Levi-Civita. At the Ukrainian Academy of Sciences, M. Kravchuk held the position of scientific secretary of UAN. He also was dean of the faculty of professional education of Kyiv University and a member of the scientific board of the City Council of Kyiv. During 1934– 38 M. Kravchuk directed the Section of Mathematical Statistics at the Institute of Mathematics of the Ukrainian Academy of Sciences.

The 1930s were the years of great terror and massive repression in USSR and particularly in Ukraine. Soviet authorities were bent on destroying the nascent Ukrainian renaissance. Kravchuk’s dedication to the development of science in Ukraine, his influence and popularity among young scientists and University students did not go unnoticed by the authorities. Mykhailo Kravchuk was arrested on the 21st of February 1938 and charged with Ukrainian nationalism, counterrevolutionary activities and espionage. These were stereotype charges ascribed to many of those who were arrested during that time. On the 28th of September, M. Kravchuk was sentenced to 20 years imprisonment, followed by additional 5 years in which he was to be deprived of all civil rights. He was sent to a concentration camp in Kolyma, remote northeast of the Soviet Union, to serve sut his sentence. There, as a prisoner, Kravchuk spend three years at hard labour. With his health ruined by the extreme conditions of hunger, cold, physical and mental deprivation, M. Kravchuk died on March 9, 1942 and was buried in the Kolyma permafrost. On September 15, 1956 the Soviet authority “rehabilitated” M. Kravchuk, since no evidence could be found that he had committed a crime. But only on the 20th of March 1992, almost 100 years after his birth, Mykhailo Kravchuk was readmitted to membership in the National Academy of Sciences of Ukraine (NANU). The same year his name was entered in the International Calendar of Scientists by UNESCO. The First Kravchuk International Conference was held at Kyiv Polytechnic Institute in 1992. Since that time there were nine such conferences. The tenth Kravchuk conference is being organized to take place in Kyiv in May 2004. Lately two books of M. Kravchuk’s works were published in Kyiv.

The first, ”Popular Scientific Works,” appeared in 2000; the second, ”Selected Mathematical Works” 792 p. (2002; Zbl 1042.01017), was published jointly with the Ukrainian Academy of Arts and Sciences in the USA. On the 16-th of May 2002, the National Technical University of Ukraine (formerly known as Kyiv Polytechnic Institute) named an auditorium in his honour, and on the 20th of May 2003, the NTUU unveiled a statue of M. Kravchuk. M. Kravchuk authored more than 180 scientific works, including 10 monographs, in various branches of mathematics: algebra, number theory, theory of functions of real and complex variables, probability, statistics and history of mathematics.

The fundamental areas of his research were: investigation of the theory of commutative matrices, quadratic and bilinear forms, linear transformations, theory of algebraic and transcendental equations; number theory; investigation of certain problems in the theory of real functions and functions of complex variable; on interpolation methods, development of the least squares method applied to the solution of differential and integral equations; the creation and mathematical proof of the generalized method of moments and its application to the approximate solutions of ordinary linear differential equations, integral equations and equation of mathematical physics (American scientists an inventor John Atanasoff employed these results in the making of the first electronic digital computer); development of correlation theory, application of the method of moments in mathematical statistics; introduction and use of polynomials associated with the binomial distribution, now known in the mathematical literature as the Kravchuk polynomials.

The present volume contains selected articles of well-known mathematicians from the US, France, Germany, Great Britain, Netherlands, Australia, Portugal, India, Japan, Russia, Ukraine and other countries, who used Kravchuk’s mathematical results and ideas to further mathematical knowledge. They are from such diverse fields as algebra, number theory, mathematical physics and coding theory. It is noteworthy, that despite the fact that Mykhailo Kravchuk’s name and works were seldom mentioned after his incarceration by the Soviet regime, his scientific ideas did reach the broad mathematical world and gave rise to further development in the field. At the end of this book is a list of publications dedicated to the development of Kravchuk’s mathematical ideas. The list is selective. While it is impossible to list all such works, the bibliography is of sufficient magnitude to allow interested scientists to become acquainted with works by Kravchuk and to advance the ideas that are Mykhailo Kravchuk’s mathematical legacy.

Contents of the book:

Parasyuk, O.; Virchenko, N. [Brief survey of the mathematical legacy of academician M. Kravchuk, p. xv-xxxv]; [List of the works of M. Kravchuk. Compilers N. Virchenko, H. Syta, xxxvi-xlviii].

Chapter 1. Algebra. Physics.

Dunkl, Charles F.; Ramirez, Donald E. [Krawtchouk polynomials and the symmetrization of hypergroups. p. 3–18, see [SIAM J. Math. Anal. 5, 351-366 (1974; Zbl 0249.43006)]];

Stanton, Dennis [Some \(q\)-Krawtchouk polynomials on Chevalley groups, p. 19–56 [Am. J. Math. 102, 625–662 (1980; Zbl 0448.33019)]];

Stanton, Dennis [Three addition theorems for some \(q\)-Krawtchouk polynomials, p. 57–79 [Geom. Dedicata 10, 403–425 (1981; Zbl 0497.43006)]];

Koornwinder, Tom H. [Krawtchouk polynomials, a unification of two different group theoretic interpretations, p. 80–92 [SIAM J. Math. Anal. 13, 1011–1023 (1982; Zbl 0505.33015)]];

Schempp, Walter [Extensions of the Heisenberg group and coaxial coupling of transverse igenmodes, p. 93–103 [Rocky Mt. J. Math. 19, No. 1, 383–394 (1989; Zbl 0723.22009)]];

Koornwinder, Tom H. [Askey-Wilson polynomials as zonal spherical functions on the \(SU(2)\) quantum group, p. 104–122 [SIAM J. Math. Anal. 24, No. 3, 795–813 (1993; Zbl 0799.33015)]];

Feinsilver, Philip; Fitzgerald, Robert [The spectrum of symmetric Krawtchouk matrices, p. 123–141 [Linear Algebra Appl. 235, 121–139 (1996; Zbl 0845.15007)]];

Ericson, Thomas; Simonis, Juriaan; Tarnanen, Hannu; Zinoviev, Victor [\(F\)-partitions of cyclic groups, p. 142–148 [Appl. Algebra Eng. Commun. Comput. 8, No. 5, 387–393 (1997; Zbl 0893.20039)]];

Zhedanov, Alexei [\(9j\)-symbols of the oscillator algebra and Krawtchouk polynomials in two variables, p. 149–165 [J. Phys. A, Math. Gen. 30, No. 23, 8337–8353 (1997; Zbl 0952.33010)]];

Roy, B.; Roy, P. [Coherent states, even and odd coherent states in a finite-dimensional Hilbert space and their properties, p. 166–176 [J. Phys. A, Math. Gen. 31, No. 4, 1307–1317 (1998; Zbl 0907.47061)]];

Hakioğlu, Tuğrul; Wolf, Kurt Bernardo [The canonical Kravchuk basis for discrete quantum mechanics, p. 177–187 [J. Phys. A, Math. Gen. 33, No. 16, 3313–3323 (2000; Zbl 1052.81553)]];

Koelink, H. T. [\(q\)-Krawtchouk polynomials as spherical functions on the Hecke algebra of type \(B\), p. 188-212 [Trans. Am. Math. Soc. 352, No. 10, 4789–4813 (2000; Zbl 0957.33014)]];

Klimyk, A. U. [Krawtchouk polynomials, \(q\)-Krawtchouk polynomials and representation theory. p. 213–229];

Atakishiyev, M. N.; Groza, V. A. [Operator approach to quantum and affine \(q\)-Krawtchouk polynomials, p. 230–241];

Savva, V. A.; Zelenkov, V. I.; Khlus, O. V. [Quantum Krawtchuouk oscillators, their dynamics under laser radiation, p. 242–258].

Chapter 2. Special functions. Mathematical physics.

Rahman, Mizan [An elementary proof of Dunkl’s addition theorem for Krawtchouk polynomials, p. 261–268 [SIAM J. Math. Anal. 10, 438–445 (1979; Zbl 0414.33007)]];

Morrison, J. A. [Weighted averages of Radon transforms on \({\mathbb Z}_ 2^ k\), p. 269–278 [SIAM J. Algebraic Discrete Methods 7, 404–413 (1986; Zbl 0587.42007)]];

Zeng, Jiang [Calcul Saalschützien des partitions et des dérangements colorés, p. 279–286 [Saalschütz calculus of partitions and coloured derangements, SIAM J. Discrete Math. 3, No. 1, 149–156 (1990; Zbl 0703.05003)]];

Zeng, Jiang [Linearization of products of polynomials of Meixner, Krawtchouk, and Charlier, p. 287–306 [SIAM J. Math. Anal. 21, No. 5, 1349–1368 (1990; Zbl 0724.33006)]];

Dette, Holger [New bounds for Hahn and Krawtchouk polynomials, p. 307-319 [SIAM J. Math. Anal. 26, No. 6, 1647–1659 (1995; Zbl 0842.33005)]];

Abdelkarim, F.; Maroni, P. [The \(D_ \omega\)-classical orthogonal polynomials, p. 320-347 [Result. Math. 32, No. 1-2, 1–28 (1997; Zbl 0889.42020)]];

Sookoo, Norris [Generalized Krawtchouk polynomials: New properties, p. 348-354 [Arch. Math., Brno 36, No. 1, 9–16 (2000; Zbl 1055.33011)]];

Yakubovich, S. [Convolution Hilbert spaces associated with the Kontorovich - Lebedev transformation. p. 355–363];

Dunkl, C. [Singular polynomials for the symmetric group and Krawtchouk polynomials. p. 363–373];

Chapter 3. Probability theory and mathematical statistics.

Hoare, M. R.; Rahman, Mizan [Cumulative Bernoulli trials and Krawtchouk processes, p. 389–415 [Stochastic Processes Appl. 16, 113–139 (1984; Zbl 0534.60060)]];

Feinsilver, Philip; Schott, René [Krawtchouk polynomials and finite probability theory, p. 416–423 [Algebraic structures and operator calculus. Vol. II: Special functions and computer science. Mathematics and its Applications (Dordrecht) 292. Dordrecht: Kluwer Academic Publishers. ix, 148 p. (1994; Zbl 1128.33300)]];

Seneta, Eugene [M. Krawtchouk (1892–1942) – Professor of mathematical statistics, p. 424-429 [Theory Stoch. Process. 3(19), No. 3-4, 388–392 (1997; Zbl 0932.01037)]];

Seneta, Eugene [Characterization of Markov chains by orthogonal polynomial systems. p. 430–433];

Korolyuk, V. S.; Makarov, V. L.; Strok, V. V. [Development of ideas of M. Krawtchouk in the theory of orthogonal polynomials. p. 434-449];

Chapter 4. Coding theory. Theory of signals. Number theory.

Odlyzko, A. M.; Sloane, N. J. A. [A theta-function identity for nonlattice packings, p. 453–457 [Stud. Sci. Math. Hung. 15, 461–465 (1980; Zbl 0471.52007)]];

Bromba, M.; Ziegler, H. [Explicit formula for filter function of maximally flat nonrecursive digital filters, p. 458–461 [Electron. Lett. 16, No. 24, 905–906 (1980)]];

Bromba, M.; Ziegler, H. [Further results on maximally flat nonrecursive digital filters, p. 462–466 [Electron. Lett. 18, No. 23, 1014–1015 (1982)]];

Camion, P.; Courteau, B.; Fournier, G.; Kanetkar, S. V. [Weight distribution of translates of linear codes and generalized Pless identities, p. 467-489 [J. Inf. Optim. Sci. 8, 1–23 (1987; Zbl 0633.94018)]];

Tolhuizen, L.; van Lint, J. H. [On the minimum distance of combinatorial codes, p. 490–493 [IEEE Trans. Inf. Theory 36, No. 4, 922–923 (1990; Zbl 0707.94014)]];

Solé, Patrick [An inversion formula for Krawtchouk polynomials with applications to coding theory, p. 494-500 [J. Inf. Optimization Sci. 11, No. 2, 207–213 (1990; Zbl 0708.05063)]];

Solé, Patrick [Packing radius, covering radius, and dual distance, p. 501–515 [IEEE Trans. Inf. Theory 41, No. 1, 268–272 (1995; Zbl 0830.94022)]];

Levenshtein, Vladimir I. [Krawtchouk polynomials and universal bounds for codes and designs in Hamming spaces, p. 516–570 [IEEE Trans. Inf. Theory 41, No. 5, 1303–1321 (1995; Zbl 0836.94025)]];

Suen, Chung-yi; Chen, Hegang; Wu, C. F. J. [Some identities on \(q^{n-m}\) designs with application to minimum aberration designs, p. 571–583 [Ann. Stat. 25, No. 3, 1176–1188 (1997; Zbl 0898.62095)]];

Habsieger, Laurent; Stanton, Dennis [More zeros of Krawtchouk polynomials, p. 584–593 [Graphs Comb. 9, No. 2, 163–172 (1993; Zbl 0805.33008)]];

Krasikov, Ilia; Litsyn, Simon [On integral zeros of Krawtchouk polynomials, p. 594–622 [J. Comb. Theory, Ser. A 74, No. 1, 71–99 (1996; Zbl 0853.33008)]];

Stroeker, Roelof J.; de Weger, Benjamin M. M. [On integral zeroes of binary Krawtchouk polynomials, p. 623–633 [Nieuw Arch. Wiskd., IV. Ser. 17, No. 2, 175–186 (1999; Zbl 1069.33010)]];

Krasikov, Ilia [Bounds for the Christoffel-Darboux kernel of the binary Krawtchouk polynomials, p. 634–639 [Barg, Alexander (ed.) et al., Codes and association schemes. DIMACS workshop, DIMACS Center, Princeton, NJ, USA, November 9-12, 1999. Providence, RI: AMS, American Mathematical Society. DIMACS, Ser. Discrete Math. Theor. Comput. Sci. 56, 193–198 (2001; Zbl 0972.33006)]];

Krasikov, Ilia; Litsyn, Simon [Survey of binary Krawtchouk polynomials, p. 640–654 [Barg, Alexander (ed.) et al., Codes and association schemes. DIMACS workshop, DIMACS Center, Princeton, NJ, USA, November 9-12, 1999. Providence, RI: AMS, American Mathematical Society. DIMACS, Ser. Discrete Math. Theor. Comput. Sci. 56, 199–211 (2001; Zbl 0981.33003)]].

Chapter 5. Method of moments.

Katchanovski, I. [Mykhailo Krawtchouk and a puzzle in the invention and patenting of the electronic computer in US. p. 655–687];

Luchka, A. Yu.; Luchka, T. F. [Mykhailo Krawtchouk and variational and projection methods. p. 688-709];

Chyp, M. M. [The moment problem and integral representation of analytic functions. p. 710–732];

Kayuk, Ya. F. [Analysis of Mykhailo Krawtchouk’s method of moments on the basis of variation correlations of the mechanics of a continuous medium. p. 733–747];

Katchanovski, I. [List of the works that develop M. Kravchuk’s mathematical ideas. p. 748–775].

Reviewer: Mikhail P. Moklyachuk (Kyïv)

### MSC:

00B60 | Collections of reprinted articles |

01A75 | Collected or selected works; reprintings or translations of classics |