Rev.Acad.Canar.Cienc., XI (Núms. 1-2), 259-275 (1999)
ANATOLII PLATONOVICH PRUDNIKOV: LIFE IN SCIENCE
Anatoly A. Kilbas
Belarusian State University, Belarus; currently at Universidad de La Laguna,
La Laguna- Tenerif e, Spain
Stefan G. Samko
Rostov State University, Russia and Universidade do Algarve, Faro, Portugal
Abstracts
The biographical data of the Russian mathematician Professor Anatolii Platonovich
Prudnikov is presented and his scientific, educational, publishing and
public activity is enlightened. A brief survey of Prudnikov's results in operational
calculus, integral transforms, special functions and partial differential
equations is given together with the list of selected publications in the chronological
order.
Mathematics Subject Class~fication: 01A70, 44A40, 46F10, 35J25, 35K05,
35K50,42A38, 33C25
Key Words and Phrases: operational calculus, integral transforms, special
functions, partial differential equations
Life and activity of A.P.Prudnikov
Anatolii Platonovich Prudnikov, the well-known authority in operational calculus,
integral transforms and special functions was born in the city of Ul'janovsk
on January 14, 1927. In 1944 he entered the Kuibyshev Aviation Institute. In
1947, after three years studies at this Institute, he continued his education as
a forth-year student of the Department of Mathematics and Mechanics in the
Kuibyshev Pedagogical Institute. Prudnikov graduated from this Institute in
1949. His professional career as a professor started from a position of Assistant
Professor at the Kuibyshev Industrial Institute. In 1952-1954 he took doctor
courses in this Institute.
In 1954 Prudnikov continued his doctor courses in Moscow at the Institute
of Precise Mechanics and Computing Technique of the Academy of Sciences of
the USSR under the supervisorship of Professor Vitalii Arsen'evich Ditkin. He
gota position of a Junior Researcher at this Institute. In 1955 he began to work
at the Computer Center of Academy of Sciences of the USSR which was organized
in this year. He worked there for more than 43 years, starting from Junior
Researcher to the head of a department. Prudnikov defended his Ph.D. dissertation
"Analytic investigations of processes in heat and mass exchanges" in 1957,
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and his Dr.Sci. dissertation "On a class of integral transforms of Volterra type
and sorne generalizations of operational calculus" in 1968 (which corresponds
to two levels of the doctor degree in the Soviet Union). In 1972 he was awarded
Professor's title, which was always considered as a title of a very high level in
the USSR.
The first publications of Prudnukov were in partial differential equations,
but he became well known due to his papers and books in operational calculus
which became the main area of his research interests for many years. This area,
closely connected with applied mathematics, was formed under the influence of
Professor Ditkin. His interests were based on both the development of theoretical
methods and applications to solution of sorne special problems of importance
in science and practice. The result of the close cooperation between Prudnikov
and Ditkin was their first book [8] published in 1958 and devoted to operational
calculus for two variables. Later investigations were summarized in the books
[29] and [30] and in the handbook [16], which were highly estimated by specialists.
In particular, Professor Prudnikov together with Professor Ditkin and
Professor Maslov ( at present the academician of Russian Academy of Sciences)
were awarded by the State Prize of the USSR in 1978.
Another area of Prudnikov's mathematical research was connected with integral
transforms and special functions. He began to work in the theory of integral
transforms together with V.A.Ditkin and continued this with his student
Yu.A.Brychkov. Their results in the theory of integral transforms of generalized
functions were presented in the book [34]. This was the first monograph
in which a survey of properties of various integral transforms in spaces of test
and generalized functions was given. Such a survey of the results in the theory
of multi-dimensional integral transforms was presented in the book [57] written
together with Brychkov, Glaeske and Vu Kim Tuan. This was also the
first monograph devoted to the theory of multi-dimensional integral transforms:
Later he was an initiator of creating five books [39], [42], [47], [55] and [56]
written together with Brychkov ap.d Marichev. These handbooks, devoted to
the evaluation of integrals and series and direct and inverse Laplace transforms
of elementary and special functions, considerably cover all known results. Unlike
other similar handbooks, they contain many new formulas and integrals of
general form which can be applied to evaluate wide classes of new integrals.
Moreover, they contain properties of many special functions which were not
mentioned earlier in monographic literature. These handbooks are very useful
not only to mathematicians, but also to specialists in physics, mechanics,
chemistry, engineering, etc who apply methods of mathematical analysis. We
also mention the book [11] of Prudnikov and Berlyand and Gavrilova where
the tables of the repeated integrals of the error functions and of the Hermite
polynomials were presented.
The next area of Prudnikov's mathematical research was concentrated in
informatics. His interests included a wide spectrum of problems in mathemati-
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cal modelling, algorithmic approach and numerical analysis in connection with
applications in computing technique. In 1992-1996 he developed methods of
solving sorne problems of mathematical modelling and numerical investigations
connected with the control of aerodynamic characteristics of flying apparatus.
Last years Prudnikov was a supervisor of investigations in medical informatics
in the State Research Experimental Institute of Aviation and Cosmic Medicine.
Under his guidance the complex of mathematical models, algorithms, programs
and nomograms was developed which was aimed to medical investigations on
estimation of a professional health of pilots and a level of radiation and electromagnetic
pollution of environments.
The results of Prudnikov's scientific activity have been published in more
than 100 articles including 10 monographs translated in England, France, Germany,
Japan, Poland and USA. His books became a table handbooks for researches
and engineers.
From the very beginning of his start at Kuibyshev Industrial Institute, Anatolii
Platonovich paid a lot of attention to mathematical education, devoted
much strength and energy to the supervision of postgraduates and trainees,
conducting seminars in the Computer Center and giving simultaneously lectures
in the All-Union Institute of Food Industry. 15 of his students defended
PhD thesis under his supervisorship and two of them defended the second-level
dissertation of Doctor of Science.
Prudnikov was widely known in Russia and other countries of the former
USSR as well as abroad. He was invited to give talks at many International
Conferences, Seminars and Workshops in the former USSR, in Bulgaria, Germany,
Great Britain, Japan, Poland, Spain and Yugoslavia. He was Visiting
Professor in Fukuoka (Japan), Jena (Germany) and Valencia (Spain), and many
times was invited by his colleagues in Finland, Germany, Great Britain, Hungary,
Poland and Yugoslavia. He actively took part in the organization of many
USSR and international conferences as a member of Organizing Committee. In
particular, he was a Vice-director of International Conference on Generalized
Functions and their Applications in Mathematical Physics held in Moscow in
1980.
Prudnikov gave a lot of his energy to publishing activity. For a long time
he was a member of the Editorial Board of the All-Union journal InzhenernoFizicheskii
Zhurnal translated in English, a member of the Publishing Council
in the Ministry of Education of the USSR, Chief of Publishing Council of Computer
Center and Editor-in-Chief of all their Proceedings. He was well known
as a good organizer and a person producing a positive impression on any partner
in negotiations. Last years Prudnikov was very busy with Editor's work
under the "Gordon & Breach Science Publishers". He was the organizer of the
international journal Integral Transforms and Special Functions which started
in 1993, and was an active Editor-of-Chief of this journal, devoting much time
to his duties in this journal till the last days. Professor Prudnikov paid a lot
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of attention to the monograph series "Analytical Methods and Special Functions"
published by "Gordon & Breach Science Publishers" , of which he was
also the Editor-in-Chief. He was also engaged into the activity connected with
the Information Bulletin Integral Transforms and Special Functions (in Russian)
prepared by the research group of the journal Integral Transforms and Special
Functions and published by the Computer Center of the Russian Academy of
Sciences.
Prudnikov took a very active part in public and scientific life in Russia
(in the USSR before) and abroad. He was a Scientific Secretary of Computer
Center and a Chief of the post-graduate courses there, a member of Directory
Board of Scientific and Methodological Council in Mathematics, Physics and
Astronomy of Russian society "Znanie", a member and Vice-Chief of Expert
Council in the Supreme Attestation Commission of the USSR and later Russia
in mathematics and mechanics. Last years he was a Chief of such an Expert
Council in the Supreme Attestation Commission of Russia and a member of
Dissertation Councils of Experts in Computer Center and in Moscow State
University. He was a member of London and American Mathematical Societies.
Prudnikov had a bright and original personality, was a joyful and witty
person, and was very communicable. Unexpected death of Professor Prudnikov
on January 10, 1999 was really a great loss for all people who collaborated with
him.
The interested reader can find sorne additional information about Prudnikov
in the Obituary
"Anatolii Platonovich Prudnikov" ,"Integral Transform and Special Functions 8
(1999), no. 1-2, to appear,
and in the paper
E.I.Moiseev, K.Skornik and W.Kierat "Life and work of Professor Anatolii
Platonovich Prudnikov (1927-1999)", Fractional Calculus and Applied
Analysis 2 (1999), no. 1, 97-106.
Brief survey of the scientific results of A.P.Prudnikov
The first investigations of Prudnikov were connected with mixed boundary
value problems for partial differential equations. In [1] he studied the system of
two parabolic equations
(1)
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a1 , b1 , a2 and b2 being constants such that
(2)
with the initial and boundary conditions u(x, y, O) = Ji (x, y), v(x, y, O) =
h(x, y), u~(O, y, t) = u;(x, O, t) = v~(O, y, t) = v~(x, O, t) = O u~(c, y, t) =
91 (y, t), u;(x, d, t) = h1 (x, t) (t?: O), v~(c, y, t) = g2(Y, t), v;(x, d, t) = h2(x , t)
(t?: O) where fk(x,y) (k = 1,2) and hi(x,t) (i = 1,2) are given functions.
Applying two-dimensional Fourier transform, he reduced this problem to a system
of ordinary differential equations and obtained the solution u(x, y, t) and
v(x, y, t) in closed form in terms of theta-functions.
In [3] Prudnikov studied two-dimensional system of the form (1)
ou 82u ov ov 82v 82u
ot = ª1 ox2 + bl ot ' ot = ª2 ox2 + b2 ox2 ' (3)
where the constants a 1 , b1 , a2 and b2 satisfy (2), with the conditions
u(x,O) = fi(x), v(x,O) = h(x) (O< x < l) (4)
u(O, t) = 91(t), v(O, t) = 92(t), u(l, t) = g3(t), v(l, t) = g4(t), (5)
ak1u(O, t) + ak2v(O, t) + ak3ux(O, t) + ak4vx(O, t) = hk(t) (k = 1, 2, 3, 4), (6)
u(O, t) = h1 (t), v(O, t) = h2(t) , u(l, t) = h3(t), v(l, t) = h4(t), (7)
where akj (l :::; k,j:::; 4) are constants and fk(x) (k = 1, 2), gk(t) (1 :::; k:::; 4)
and hk(t) (1:::; k:::; 4) are given functions for x E (O, l) and t E (O, oo). Applying
to (3)-(6) the double mixed Fourier and Laplace transform, he represented the
solutions u(x, t) and v(x, t) as sums of integrals containing unknown functions
which are determined form the boundary condition (7) through a system of
Volterra integral equations.
In [3] Prudnikov showed sorne application of the problem (4)-(7) in the
thermo-diffusion theory. His papers [2], [4]-[7] and [9]-[10] were devoted to
applications of partial differential equations to investigation of other applied
problems. In [2] he dealt with a solution of sorne problems in the theory of
molecular transfer. Sorne study of certain processes of heat and mass transfer
was given in [4]-[7]. Sorne problems in the theory of filtration of fluid were considered
in [9] and a problem in the theory of heat conduction was discussed in
[10].
The above applied results led Prudnikov to the main area of his research in
the field of operational calculus, in which he had a close cooperation with Ditkin.
Ditkin and Prudnikov developed the operational calculus by Heavisade and by
Mikusinski. In [8] they extended the investigations by Voelker and Doetsch
[Die zweidimensionale Laplace-Transformation, Verlag Brkhauser, Basel, 1950,
MR 12, 699], concerning the properties of two-dimensional Laplace transform,
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and constructed the operational calculus in two variables on the basis of the
two-dimensional Laplace convolution. These investigations were generalized in
[30] where they developed the operational calculus based on the Volterra-type
two-dimensional transform
a2 r r F@G=axay}o lo F(x-t,y-s)G(t,s)dtds. (8)
In [29] using the idea of the generalized Laplace transform, Ditkin and
Prudnikov extended Mikusinski's operational calculus to the complex domain.
They showed that the set of all Laplace transformable operators forros a field
Mo e M, where M is the Mikusinski's quotient field. This field is isomorphic
to the field Mo of all functions of a complex variable represented in the
forro F*(z)/G*(z) where F*(z) and G*(z) are the Laplace integrals off and
g, respectively, in sorne half-plane Re(z) > e, e is real constant. Ditkin and
Prudnikov studied the representation of the whole field M of Mikusinski's operators
by functions of a complex variable and proved that M is isomorphic to
the field M which consists of elements closely related to functions of the forro
F* (z) /G* (z).
Prudnikov paid a lot of attention to operational calculus of Bessel type
operators. Ditkin first constructed operational calculus for the Bessel operator
B2 = DxxDx , Dx = d/dx , by defining ring MB2 with the convolution
F@G = B2 lx dt 11 F(ty)G[(x - t)(l - y)Jdy , (9)
as a multiplication of the elements F and G of MB2 • Developing the results of
Ditkin and Meller who studied an operational calculus for the operators closely
connected with B2, in [12] Prudnikov investigated the operational calculus for
the operator B3 = DxxDxxDx. He defined a ring MB3 with the convolution
product
F@G = B31x 11 dy 11 F(tys)G[(x - t)(l - y)(l - s)Jds (10)
and showed that in such an operational calculus the integral transform
2100 M(xt)w(t)dt, ¡00 dt
M(u) = 2 exp (-4u/t2) K0 (t)-,
o t
(11)
with McDonald function K 0 (z) plays the same role as the Laplace transform in
Mikusinski's operational calculus. The results obtained were extended by Ditkin
and Prudnikov to the Bessel operators Bk = (1/x)(xD)k (k = 2,3, · · ·), see [13],
where application to the differential equation B3y +>.y = O was also considered.
In [14] they gave a Mikusinsk1-type treatment of an operational calculus based
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on the operator B2 and indicated that such a calculus is related to the Meijer
integral transforrn
(Kow)(x) = 2 fo00 Ko (2(xt) 1! 2 ) w(t)dt (12)
in the sarne way as the classical operational calculus with the Laplace transforrn.
Ditkin and Prudnikov also considered discrete analogies of the constructions
above and developed the operational calculus of functions of one and two integer
argurnents in [15] and [18], respectively, with indication of applications in discrete
analysis. In [19] they suggested the method of inclusion of non-integrable
functions into operational calculus based on Hadamard's ideas on finite parts of
divergent integrals. In [26]-[27] they developed the operational calculus based
on the generalization of the relations (9) by
1 1x F \¿9 G = u-1 - F(x - t)G(t)w(x - t)w(t)dt
w o
(13)
and
1 ¡x 11 F\¿9G = u-1 - dt F(ty)G[(x-t)(l-y)Jw[(x-t)(l-y)Jw(ty)dy, (14)
w o o
respectively, where u-1 is the operator inverse to a certain linear operator U and
w(x) is a function for which there exists the Laplace transform in the forrner
case and the Meijer transforrn (12) in the latter. We also note Prudnikov's
papers [24] and [25] where sorne problems in the theory of operational calculus
were discussed.
In [28] Ditkin and Prudnikov suggested a general rnethod for developing an
operational calculus on the basis of a generalization of (13) and (14) in the form
F\¿9G = u-1 (FG). (15)
Here u- 1 is the operator inverse to the linear operator U defined by Uf= lof for
a function fin a cornmutative ring L with a multiplication f g = w-1 (wf * wg),
w f * wg being the Laplace convolution, w is a linear operator defined on L with
values in a set of functions f defined on (O, oo) and Lebesgue integrable on any
interval (O, A), and F and G are elements of the ideal of the ring L generated
by a fixed element lo E L.
The last interest of Prudnikov in operational calculus was connected with
perfect operators, and his investigations with Ryabtsev in this field were presented
in two monographs [53] and [54]. The first one deals with a rigorous
exposition of the basis and principal results of the theory of operational calculus
of perfect operators. The whole construction is based on distribution theory,
and the perfect operators are introduced as operators in a commutative algebra
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without divisors of zero. Among the discussed problems there one can find an
algebraization of the initial system with respect to a linear operator, the fundamental
formula of the operational calculus with applications to differential
and integro-differential equations, theory and applications of systems that are
left compact supported, the convergence of sequences and series of perfect operators
and the continuity properties of the corresponding algebraic operations,
the infinitesimal properties of the ring of perfect abstract operators. The second
book [2] is devoted to differentiation and integration of operator functions and
to the method of superposition. The definitions of differentiation and integration
with respect to one and two real parameters are given, theorems concerning
ordinary and partial linear differential equations are proved with applications
to solving, in a purely algebraic way, the wave, heat and telegraph equations of
two variables and equations with delay.
In [61] Prudnikov presented a comprehensive survey of latest results in the
theory of operational calculus. This paper can be considered as a continuation
of the survey articles [17] and [22] written together with Ditkin and the
survey paper [35] with Brychkov and Shilov. Beginning from a short historical
comment on the Heaviside classical operational calculus, he gave Plesner 's rigorous
description of the operational calculus based on the theory of operators
developed by Mikusinski, characterized Mikusinski operational calculus and its
development by Ditkin and Prudnikov, reviewed the operational calculus based
on the Schwartz theory of distribution and on Antosik's sequential approach
to distributions, paid the special attention to the convolution calculus studied
by Dimovski and presented his and Ryabtsev operational calculus of perfect
operators. Prudnikov indicated general applications of operational calculus to
nonlinear systems of automatic control, linear escape problem, and linear problem
of tracking motion. He also formulated a new problem for four nonlinear
partial differential equations, and suggested that these equations need solutions
based on the new convolution in the construction of an operational calculus
as a branch of nonlinear functional analysis. This Prudnikov's article is very
informative and, surely, it will provide new stimulating interest in the advanced
study and research in operational calculus.
The above Prudnikov's investigations in operational calculus were always
connected with problems in applications. In the eighties he paid a special attention
to two-dimensional boundary value problems for the Laplace and the
Poisson equations. In [40] Prudnikov together with his student Vlasov considered
the boundary value problems for the Laplace equation in a class of
two-dimensional plane domains with complicated boundary, often arising in
physics and mechanics, with a non-homogeneous Dirichlet condition on a part
of boundary and the homogeneous Dirichlet or Neumann condition on the rest
of boundary. They developed the method for solution of this problem based
on the representation of the solution in the form of a series in powers of a
conformal mapping of an extension of the original domain. They considered a
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generalization of such a method in [43], and in [49] applied this method to solve
sorne boundary value problems for the Poisson equation in the above domains.
Vlasov, Volkov and Prudnikov in [48] considered the Dirichlet problem for the
Laplace equation in a disk with a round angular notch and gave the solution in
closed form that is infinitely differentiable in the domain and on the contour of
the rounded notch. In [44] and [45] they investigated the Dirichlet problem for
L-type domain and for a circle with two notches, respectively. In [46] Vlasov,
Volkov, Prudnikov and Yakovleva obtained the so called fundamental frequencies
and the eigen-functions for the torsional oscillation, when the cross-section
of the cylindrical shaft is a clise with a notch that is bounded by a rounded
corner.
Numerical examples for the solution of the Dirichlet problem for the Laplace
equation in considered domain were also presented in [48]. These investigations
were continued by Vlasov, Volkov, Prudnikov, Yakovleva and Vladimirova
in [51]. They gave a description of a complex of algorithms and programs
for the solution of the Dirichlet problem for the Poisson equation in domains
whose boundary is a polygon of an arbitrary form, one of the angles of which is
smoothly rounded-off by an are of a circle. The developed algorithms allowed
to obtain both the solution of the boundary value problem itself and its derivatives
in the domain and its boundary with high degree of accuracy. We also
note that earlier Prudnikov together with Zurina and Popova applied the Pade
approximate methods to numerical solution of the integral equations of transfer
theory and quantum mechanics in [31] and [32], respectively, see also [33] in this
connection.
Prudnikov's interest to the theory of integral transforms was caused by an
operational calculus, see (11) and (12), and applied problems. His results in
this field were presented in the book [29] and the survey articles [20] and [23]
written together with Ditkin. We also note the book [34] and the review paper
[41] written with Brychkov devoted to the theory of integral transforms of
generalized functions, and the monograph [57] with Brychkov, Glaeske and Vu
Kim Tuan dealt with multidimensional integral transforms. The special interest
of Prudnikov was connected with the Watson transform. Such a transform was
introduced by Watson as the generalization of the Fourier cosine transform Fcf
given by
(Fcf)(x) = (~) i¡2 ..!!:.._ {oo sin(xt) J(t)dt
7r dx }0 t
(16)
by replacing the kernel (2/n) 112 sinx/x by the function 'lf;(x). The Watson
transform can be deined by
(W J)(x) = d~ ( x fo00 'lf;(xt)f(t)dt) (17)
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where the function 'lfJ(x) satisfies the condition
¡00 { o,
'lfJ(xt)'ljJ(yt)dt =
-oo 1/[rnax(lxl, lyl)]
if xy <O,
(18)
if xy > O.
Prudnikov investigated the properties of such an operator in L2(-oo, oo) in [52]
and together with Skornik in [64]. In particular, it was proved that the Watson
operator W can be represented in a unique way as a series of the operators of
the forrn (TS)nT and S(TS)n (n =O, 1, 2, · · ·), where
d ( rl/x )
(Tf)(x) = dx x Jo f(t)dt , (Sf)(x) = ~! (~) . (19)
Another generalizations of cosine- and sine-transforrns given by the integral
transforrn with confluent hypergeornetric functions as kernels, were investigated
by Prudnikov together with Moiseev and Skornik in [63].
Prudnikov carne to the Watson transforrn while studying in [36]-[37] certain
problerns of the theory of heat conduction. He reduced these problerns to the
heat equation
(20)
with the boundary conditions Ux + aulx=li+bt = h1(t), ulx=l2+bt = h2(t), >. and
a being real nurnbers and h;(t) (i = 1, 2) given functions, and with the initial
condition ult=-oo = O, and obtained the explicit solution u(x, t) of this problern
in terrns of the Watson transforrn (17) with sorne special 'lfJ(x). Prudnikov and
Bartoshevich [38] solved sorne problerns of rnathernatical physics by using the
rnethod of decornposition of certain integral operators into orthogonal Watson
operators.
The rnain interests of Prudnikov in the theory of special functions were
connected with calculation of integrals of special functions. In this connection
the reader rnay be referred to the books [39], [42], [47], [54], [55] and a survey
article [50] written together with Brychkov and Marichev. In the nineties he
especially interested in problerns concerning orthogonal systerns of polynornials
and special functions. His first results in this direction were obtained in [21]
where he gave the explicit expressions for the series
00 ~ >.n p (t)P (x) ~ Ln(t)Ln(x)
~ n n ' ~ n+l/2
L An Ln(t)Pn(x ), (21)
n=O
with Laguerre Ln(t) and Legendre Pn(x) polynornials, in terrns of the Bessel
function of the first kind J 0 (z), the rnodified Bessel function I 0 (z) and McDonald
function Ko(z) .
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Prudnikov carried out investigations in this area together with Moiseev. In
[59] they introduced a new system of orthonormal functions defined in terms of
the Gauss hypergeometric function 2F1(a, b;; z) by
[ a (1 ix)] (1 ix)-l-a/2
p~(x, a)= A~ 2F1 -n, 1 + 2; a+ l; 2 + 2a 2 + 2ª , (22)
where
A"'-f(l+a/2) (r(a+n+l)) -oo<x<oo, n=±l,±2,···. (23)
n - 2f(l +a) n!?ra '
When a= O anda= 1, the system (22) is reduced to the the orthogonal system
= o - -1/2 (ix - l)n
p(x)-Pn(x,1)-7r (ix+l)n+l' -oo<x<oo, n=±l,±2,···, (24)
introduced by Wiener.
Moiseev and Prudnikov [59] proved that p~(x, a) is a complete orthonormal
system in L 2 (0,oo) as well as the systems J2C;::(x,a) and J2S;::(x,a) of real
functions c;::(x,a) and s;::(x,a) defined via (22) by
C~(x, a)+ iS~(x, a)= \i2p~(x, a). (25)
They also constructed the corresponding differential equation and the generating
function for p~(x, a). These investigations were continued in [60] where
the representation for the Wiener function p(x) in (24) via Chebyshev polynomials
of the first and second kind T2n+l ((1 + t 2 ) 112 ) and U2n+l ((1 + t 2 ) 112 )
and the differentiation relation between McDonald function K n+ 1 ¡2 ( x) and Laguerre
polynomial Ln(x) were applied to construct several complete orthogonal
systems of functions in L2 (0, oo) and L2 (0, 1). In [62] Moiseev and Prudnikov
investigated the system sin[(n - /3/2)B] (n = 1, 2, · · ·) of sines and the system
1, cos[(n - f3/2)B] (n = 1,2,···) of cosines with real f3 on the interval
[O, 7r]. They proved that these systems form a basis in sorne subspaces of the
Sobolev space W~(O, 7r) and the corresponding series are convergent, provided
that p E (1, oo) and f3 satisfy additional conditions. They also obtained similar
results for the associated system of exponential functions exp[i(n -f3/2)sgn(n)]
(n =O, ±1, ±2, · · ·) in a subspace of W~(-7r, 7r).
In [58] Prudnikov considered the orthonormal system ofpolynomials V0 (x, k) =
1, Vi (x, k), · · ·, Vn(x, k) with k >O on the interval [O, oo) such that
('° { O, if m i- n,
Jo Vn(x,k)Vm(x,k)((x,k)dx= Dmn =
o 1 if m = n,
(26)
with respect to the weight function ((x, k) defined by the Mellin-Barnes integral
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x >O, a> O, Re(s) >O. (27)
The system Vn ( x, k) is a generalization of the system of Laguerre polynomials
(-l)n Ln(x) with respect to the weight ((x, 1) = e-x which corresponds to
k = l. When k = 2, the relations (26) and (27) give a new system of orthogonal
polynomials Vn(x) = Vn,2(x) with respect to the weight ((x, 2) = 2Ko( fa).
Prudnikov formulated the problem to find the generating function, an analogue
of Rodrigues' formula, the recurrence relation anda differential equation for the
orthogonal polynomials Vn(x) and for more general polynomials Vn ,k(x) with
k > 2. He discussed the recurrent formula for Vn(x) , and found such a recurrent
relation for the weight function 2fx[K0 ([xl) on the whole axis (-oo, oo) and for
the corresponding orthogonal polynomials Wn(x) symmetric with respect to the
origin.
The above problems as well as other problems and ideas by Professor Anatolii
Platonovich Prudnikov will stimulate both the further development of methods
of operational calculus, integral transforms and special functions and their applications
to applied problems in physics, mechanics and other sciences.
References
[1] Prudnikov, A.P. A solution of a mixed problem in the integral form for
a system of two parabolic differential equations (Russian). Dokl. Akad. Nauk
SSSR (N.S.) 115 (1957), no. 5, 869-871. (Reviews in Mathematical Reviews
MR 20 ~1848; Zentralblatt für Mathematic Zbl.81,92).
[2] Prudnikov, A.P. A solution of sorne problems in the theory of molecular
transfer for solids of the simplest geometric form (Russian). lzv. AN SSSR.
Otd. Tehn. Nauk 1957, No 8, 143-145.
[3] Prudnikov, A.P. The solution of a mixed boundary problem in the thermodiffusion
theory (Russian). Dokl. Akad. Nauk SSSR 119 (1958), no. 2,
249-251 (MR 20 "5359; Zbl.83,91).
[4] Prudnikov, A.P. To an investigation of heat and mass transfer in dispersion
surroundings (Russian). Inzh.-Fiz. Zh. 1 (1958), no. 4, 81-86.
[5] Prudnikov, A.P. Analytic investigation of processes of heat and mass
transfer in convective drying (Russian). Izv. Akad. Nauk SSSR. Otd. Tehn.
Nauk 1958, No. 10, 63-67 (MR 20 "7512; Zbl.112,416).
[6] Lykov, A,.V.; Prudnikov, A.P. Toan investigation of phenomena of transfer
of the heat and substance in heavy surroundings (Russian). Dokl. AN BSSR.
2 (1958), no. 8, 334-337.
[7] Prudnikov, A.P. Toan investigation of processes of heat and mass transfer
while heating and drying a fuel in energo-technological installations (Russian).
Izv. AN BSSR: Ser. Fiz.-Tehn. Nauk, 1958, No. 3, 11-23.
270
© Del documento, de los autores. Digitalización realizada por ULPGC. Biblioteca Universitaria, 2017
[8] Ditkin, V.A.; Prudnikov, A.P. Operational Calculus in Two Variables and
its Applications (Russian). Gosudarstv. Izdat. Fiz.-Mat. Lit., Moscow, 1958,
178 pp. (MR 22 U881; Zbl.126,314). English publication in International Series
of Monographs on Pure and Applied Mathernatics, Vol. 24. Pergarnon Press,
New York-Oxford-London-Paris, 1962, 167 pp. (MR 25 U2397; Zbl.116,309).
[9] Prudnikov, A.P. To the theory of filtration of fluid in the grounds (Russian).
Izv. AN SSSR. Energetika i Avtomatika, 1959, No. 1, 119-121.
[10] Prudnikov, A.P. On a problern in the theory of heat conduction (Russian).
Inzh.-Fiz. Zh. 3 (1960), no. 5, 136-137.
[11] Berlyand, O.S.; Gavrilova, R.I.; Prudnikov, A.P. Tables of integral error
functions and Hermite polynomials (Russian). Izd. AN BSSR, Minsk, 1961,
164 pp English publication in Pergarnon Press book, The Macrnillan Co., New
York, 1962, 163 pp. (MR 27 U5937; Zbl.105,113).
[12] Prudnikov, A.P. On the theory of operational calculus (Russian). Dokl.
Akad. Nauk SSSR 142 (1962), no. 4, 794-797; English transl. in Soviet Math.,
Doklady 3 (1962), 172-176 (MR 24 UA3481; Zbl.118,316).
[13] Ditkin, V.A.; Prudnikov, A.P. Operational calculus of Bessel operators
(Russian). Z. Vychisl. Mat. i Mat. Fiz. 2 (1962), no. 6, 997-1018; English
transl. in USSR Comput. Math. Math. Phys. (1964) , 1184-1212 (MR 26 U6704;
Zbl.141,114).
[14] Ditkin, V.A.; Prudnikov, A.P. On the theory of operational calculus
sternrning frorn the Bessel equation (Russian). Z. Vychisl. Mat. i Mat. Fiz. 2
(1963), no. 3, 223-238; English transl. in USSR Comput. Math. Math. Phys.
3 (1963), 296-315 (MR 27 U2817; Zbl.142,396).
[15] Ditkin, V.A.; Prudnikov, A.P. On operational calculus of functions of
integer argurnent and sorne of their aplications in discrete analysis (Russian).
Inzh.-Fiz. Zh. 7 (1964), no. 7, 101-115 (Zbl.141,115).
[16] Ditkin, V.A.; Prudnikov, A.P. Handbook on Operational Calculus (Russian).
Vysshaja Shkola, Moscow, 1965, 466 pp. (Zbl.126,314). French translation
in Formulaire pour le calcul opérationnel (French). Masson e Cie, editurs,
París, 1967, 468 pp. (MR 36 U1931; Zbl.146,362).
[17] Ditkin, V.A.; Prudnikov, A.P. Operational calculus. (Russian) Mathematical
Analysis, 1964 (Russian), pp. 7-75; Akad. Nauk SSSR Inst. Nauchn.
Inforrnacii, Moscow, 1966 (MR 33 U4610).
[18] Ditkin, V.A.; Prudnikov, A.P. Operational calculus of functions of two
integral argurnents and sorne of their applications (Russian). Inzh.-Fiz. Zh. 17
(1967), no. 4, 697-708.
[19] Ditkin, V.A.; Prudnikov, A.P. A rnethod for introducing the finite
parts of divergent integrals, and applications of thern in operational calculus
(Russian). Differentsial'nye Uravnenija 3 (1967), no. 10, 1772-1781 (MR 36
U4282; Zbl.155,450); English transl. in Differenial Equations 3 (1967), 922-926.
271
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(Zbl.233.44006).
[20] Ditkin, V.A.; Prudnikov, A.P. Integral transfroms (Russian). Mathematical
Analysis, 1966 (Russian), pp. 7-82; Akad. Nauk SSSR Inst. Nauchn.
lnformacii, Moscow, 1967; English transl. in Progress Math. 4 (1969), 1-85
(MR 58 U2037; Zbl.197,379).
[21] Prudnikov, A.P. On bilinear expansions containing Legendre polynomials
(Russian). Dokl. AN BSSR. 12 (1968), no. 2, 119-122. (Zbl.203,62).
[22] Ditkin, V.A.; Prudnikov, A.P. Operational Calculus. Progress in Mathematics
1 (1968), Plenum Press, New York, 1-74.
[23] Ditkin, V.A.; Prudnikov, A.P. Integral Transforms. Progress in Mathematics,
4 (1969), Plenum Press, New York, 1-85 (Zbl.197,379)
[24] Prudnikov, A.P. To the theory of operational calculus (Russian). Dokl.
AN BSSR. 13 (1969), no. 3, 222-224 (Zbl.179,201).
[25] Prudnikov, A.P. To the theory of operational calculus (Russian). Problems
of Heat and Mass Transfers, Energija, Moscow, 1970, 317-322.
[26] Prudnikov, A.P. Operational calculus of a class of generalized Bessel
operators. Symposium on Operational Calculus and Generalized Functions,
Dubrovnik, Yugoslavija, 1971, 3-11.
[27] Ditkin, V.A.; Prudnikov, A.P. Operational calculus of certain differential
operators (Russian). Problems in applied mathematics and mechanics
(dedicated to A. A. Dorodnicyn on his si:Itieth birthday) (Russian), pp. 75-85,
Izdat. Nauka, Moscow, 1971 (MR 58 Ul 7702; Zbl.277.44008).
[28] Ditkin, V.A.; Prudnikov, A.P. One method of constructing an operational
calculus (Russian). Inzh.-Fiz. Zh. 24 (1973), no. 6, 1114-1117; English
transl. in J. Engrg. Phys. 24 (1975), no. 6, 778-781. (MR 52 ijll95).
[29] Ditkin, V.A.; Prudnikov, A.P. Integral transforms and operational calculus,
Second edition. Mathematical Reference Library, Vol.5. Izdat. Nauka,
Moscow, 1974, 542 pp. (MR 58 U29867a; Zbl.298.44007). The first edition in Gosudarstv.
Izdát. Fiz.-Mat. Lit., Moscow, 1961 (Zbl.99,89). French translation
in Transformations intégrales et calcul opérationnel (French). Second edition.
Mir, Moscow, 1982, 435 pp. (MR 85a:44001; Zbl.494.44001) .
[30] Ditkin, V.A.; Prudnikov, A.P. Operational calculus, Third edition. Izdat.
Vysshaja Shkola, Moscow, 1975, 467 pp. The first edition in Vysshaja
Shkola, Moscow, 1965 (Zbl.126,314). French translation in Calcul opérationnel
(French). Second edition. Mir, Moscow, 1983, 439 pp. (MR 85a:44002)
[31] Zurina, M.I; Popova, A.M.; Prudnikov, A.P. On an approximate method
for solving the integral equation of transfer theory (Russian). Inzh.-Fiz. Zh. 31
(1976), no. 1, 111-115.
[32] Zurina, M.I; Popova, A.M.; Prudnikov, A.P. Nume.rical solution of an
integral equation of quantum mechaniccs by Pado's method (Russian). Con-
272
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tinue Fractions and their Applications 1976; Institute of Mathernatics of AN
UkrSSR, Kiev, 51-52.
[33] Zurina, M.I; Popova, A.M.; Prudnikov, A.P. A class of rnultiple integrals
of transfer theory (Russian). Inzh.-Fiz. Zh. 31 (1976), no. 3, 550-553.
[34] Brychkov, Yu.A.; Prudnikov, A.P. Integral Transforms of Generalized
Functions. Mathernatical Reference Library, Izdat. Nauka, Moscow, 1977, 287
pp. (MR 58 U2036; Zbl.464.46039). English translation in Gordon and Breach
Science Publishers, New York, 1989, 343 pp. (MR 91e:44001; Zbl. 729.46016).
[35] Brychkov, Yu.A.; Prudnikov, A.P.; Shilov, V.S. Operational calculus
(Russian). Mathematical analysis, Vol. 16 (Russian), pp. 99-148; VINITI,
Moscow, 1979; English transl. in J. Sov. Math. 15 (1981), 733-765 (MR
80g:44010; Zbl.407.44002 and 451.44003).
[36] Prudnikov, A.P. Applicaiton of Watson operator to the solution of certain
problerns of the theory of heat conduction. (Russian) Inzh.-Fiz. Zh. 37
(1979), no. 3, 503-507
[37] Prudnikov, A.P. Watson operators and their application to the solution
of certain problerns of the theory of heat conduction (Russian). Seminar on
Numerical Methods far Solving Balance Equations (Berlín, 1980) (Russian),
pp. 79-82; Rep. 1980, 5, Akad. Wiss. DDR, Berlin, 1980 (MR 82a:35052; ;
Zbl.443.35036).
[38] Bartoshevich, M.A.; Prudnikov, A.P. Watson operators and sorne of
their applications (Russian). Generalized functions and their applications in
mathematical physics, Proc. Intern. Conf., M., 1980, 430-434 (Zbl.517.44007).
[39] Prudnikov, A.P.; Brychkov, Yu.A.; Marichev, O.I. Integrals and Series.
Elementary Functions (Russian). Nauka, Moscow, 1981, 799 pp. (MR
83b:00009; Zbl.511. 00044). English translation in Gordon and Breach Science
Publishers, New York, 1986, 798 pp. (MR 88f:00013; Zbl.733.00004). Japanese
translation in Maruzen Co., Tokyo, 1991, 799 pp.
[40] Vlasov, V.I. ; Prudnikov, A.P. Asyrnptotic behavior of the solutions of
sorne boundary value problerns for the Laplace equation in the case of deforrnation
of the dornain (Russian). Current problems in mathematics, Vol. 20, 3-36;
Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn.
Inforrn., Moscow, 1982; English translation in J. Soviet Math. 25 (1984), no.
5, 1351-1379. (MR 85c:31004; ; Zbl.523.35028 and 544.35029).
[41] Prudnikov, A.P.; Brychkov, Yu.A. Integral transforrnations of generalized
functions (Russian). Mathematical analysis, Vol. 20 (Russian), pp. 78-115;
Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn.
Inforrnatsii, Moscow, 1982; English transl. in J. Sov. Math. 34 (1986), 1630-
1655. (MR 84g:46062; ; Zbl.552.46021 and 595.46043).
[42] Prudnikov, A.P.; Brychkov, Yu.A.; Marichev, O.I. Integrals and Series.
Special Functions (Russian). Nauka, Moscow, 1983, 751 pp. (MR 85b:33002;
273
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Zbl.626. 00033). English translation in Gordon and Breach Science Publishers,
New York, 1986, 750 pp. (MR 88f:00014 and 89i:00030; Zbl.733.00005).
Japanese translation in Maruzen Co., Tokyo, 1992, 751 pp.
[43] Vlasov, V.I.; Prudnikov, A.P. A method for solving boundary value problems
for the Laplace equation in domains with a curvilinear boundary (Russian).
Second international symposium on complex analysis and applications (Budva,
1986); Mat. Vesnik 38 (1986), no. 4, 617-623 (MR 88i:31004; Zbl.617.31001).
[44] Vlasov, V.I.; Volkov, D.V.; Prudnikov, A.P. The Dirichlet problem for
L-type domain. Complex analysis and applications '85 (Varna, 1985), 743-754,
Bulg. Acad. Sci., Sofia, 1986 (MR 88m:31003a; Zbl.628.35022) .
[45] Vlasov, V.I.; Volkov, D.V.; Prudnikov, A.P. The Dirichlet problem for a
circle with two notches. Complex analysis and applications '85 (Varna, 1985),
755-761, Bulg. Acad. Sci., Sofia, 1986 (MR 88m:31003b).
[46] Vlasov, V.I.; Volkov, D.V.; Prudnikov, A.P.; Yakovleva, E.A. Torsional
oscillations of the shaft with longitudinal notch Complex analysis and applications
'85 (Varna, 1985), 762-766, Bulg. Acad. Sci., Sofia, 1986 (Zbl.619.73048).
[47] Prudnikov, A.P.; Brychkov, Yu.A. ; Marichev, 0.I. Integrals and Series.
Supplementary Chapters (Russian). Nauka, Moscow, 1986, 800 pp. (MR
88f:00012; Zbl.606. 33001). English translation in Integrals and Series. Vol.
3. More Special Functions. Gordon and Breach Science Publishers, New York,
1990, 800 pp. (MR 91c:33001).
[48] Vlasov, V.I.; Volkov, D.V.; Prudnikov, A.P. Solution of the Dirichlet
problem in a domain with a boundary containing a rounded comer (Russian).
Z. Anal. Anwendungen 6 (1987), no. 1, 61-73 (MR 88h:35029; Zbl.645.31001) .
[49] Vlasov, V.I.; Prudnikov, A.P. Solution of sorne boundary value problems
in domains with a curvilinear boundary. (Russian) Problems in applied mathematics
and information sciences {Russian), 169-186, Nauka, Moscow, 1987 (MR
89c:73014; Zbl.685.35034).
[50] Prudnikov, A.P.; Brychkov, Yu.A.; Marichev, O.I. Calculation of integrals
and the Mellin transform (Russian). Mathematical analysis, Vol. 27
(Russian), 3-146; Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst.
Nauchn. i Tekhn. Inform., Moscow, 1989; English transl. in J. Sov. Math. 54
(1991), no. 6, 1239-1341 (MR 90m:44010; Zbl.725.44001).
[51] Vlasov, V.I.; Volkov, D.V.; Prudnikov, A.P.; Yakovleva, E.A.; Vladimirova,
N.Yu. A system of algorithms and programs for solving a class of boundary value
problems (Russian). Analytical and numerical methods far solving problems in
mathematical physics (Russian), 71-80, Acad. Nauk SSSR, Vychisl. Tsentr,
Moscow, 1989.
[52] Prudnikov, A.P. On Watson transform Fractional Calculus and its Applications,
Proc. Intern. Conf. (Tokyo, 1989), 185-190; College of Engineering,
Nihon University, 1990 (Zbl.751.44004).
274
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[53] Prudnikov, A.P.; Ryabtsev, I.I. Operational calculus of perfect operators
(Rusian). Acad. Nauk SSSR, Vychisl. Tsentr, Moscow, 1991, 126 pp. (MR
93h:44008; Zbl.852.44006).
[54] Prudnikov, A.P.; Ryabtsev, I.I. Operational calculus of perfect operators
(special sections) Rusian). Ross. Acad. Nauk, Vychisl. Tsentr, Moscow, 1992,
99 pp. (MR 95c:44003; Zbl.816.4002) .
[55] Prudnikov, A.P.; Brychkov, Yu.A. ; Marichev, O.I. Integrals and Series.
Vol. 4. Direct Laplace Transforms. Gordon and Breach Science Publishers,
New York, 1992, 619 pp. (MR 93c:44003; Zbl.786.44003) .
[56] Prudnikov, A.P.; Brychkov, Yu.A.; Marichev, O.I. Integrals and Series.
Vol. 5. Inverse Laplace Transforms. Gordon and Breach Science Publishers,
New York, 1992, 595 pp. (MR 93i:44001; Zbl.781.44002).
[57] Brychkov, Yu.A.; Glaeske, H.-J .; Prudnikov, A.P.; Vu Kim Tuan Multidimensional
Integral Transformations. Gordon and Breach Science Publishers,
Philadelphia, 1992, 386 pp. (MR 93j:44002; Zbl.752.44004) .
[58] Prudnikov, A.P. Orthogonal polynomials with ultra-exponential weight
functions J. Comput. Appl. Math. 48 (1993), 239-241.
[59] Moiseev, E.I.; Prudnikov, A.P. On a complete orthogonal system of
special functions. Proceedings of Seventh Spanish Symposium on Orthogonal
Polynomials and Applications (VII SPOA) (Granada, 1991). J. Comput. Appl.
Math. 49 (1993), no. 1-3, 201-206 (MR 94m:42057; Zbl.794.42020).
[60] Moiseev, E.I.; Prudnikov, A.P. The Wiener orthogonal system and related
orthogonal polynomials. Bull. Polish Acad. Sci. Math. 42 (1994), no. 2,
141-148 (Zbl.818.42014).
[61] Prudnikov, A.P. On the continuation of the ideas of Heaviside and
Mikusinskii in operational calculus. Different aspects of differentiability (Warsaw,
1993), Dissertationes Math. (Rozprawy Mat.) 340 (1995), 237-287. (MR
96f:46075; Zbl.851.44005).
[62] Moiseev, E.I.; Prudnikov, A.P. On the basis property of systems of sines
and cosines in the Sobolev spaces Bull. Polish Acad. Sci. Math. 44 (1996), no.
4, 401-409 (MR 97m:46053; Zbl.868.46022) .
[63] Moiseev, E.I.; Prudnikov, A.P.; Skornik, U. On the generalized Fourier
sine- and cosine-transforms Integral Transform. Spec. Funct. 7 (1998), no. 1-2,
163-166.
[64] Prudnikov, A.P.; Skornik, U. A remark on Watson transform Transform
Methods and Special Functions. Varna '96, Ed. P.Rusev, I.Dimovski,
V.Kiryakova, 360-367. Bulgarian Academy of Sciences, Sofia, 1998.
275
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