Correlations between elements and sequences in a numerical prism Текст научной статьи по специальности «Математика»

DOI: 10.14529/mmph190104

CORRELATIONS BETWEEN ELEMENTS AND SEQUENCES IN A NUMERICAL PRISM

M.S. Tokmachev

Yaroslav-the-Wise Novgorod State University, Veliky Novgorod, Russian Federation E-mail: [email protected]

A numerical prism, previously introduced by the author as an ordered set regarding the research of a three-parameter probability distribution of the hyperbolic-cosine type, which is a generalization of the known two-parameter Meixner distribution, is being considered. In geometry-related terminology, elements of a numerical prism are the coefficients of moment-forming polynomials for the specified distribution, which are obtained with the use of both differential and algebraic recurrence correlations.

Each one of the infinite number of elements depends on three indices determining its position in a prism. Fixation of one or two indices results in cross-sections of the prism, which are numerical triangles or sequences. Among them, there are such well-known cross-sections as the Stirling number triangle, number triangle of coefficients in the Bessel polynomials, sequences of tangent and secant numbers, and others. However, the majority of numerical sets in the prism's cross-sections have never been described in literature before.

Considering the structure and construction algorithm, cross-sections of the numerical prism turn out to be interconnected not only by the general construction formula but also by certain correlations. As a result, formulas of connection between various groups of elements are presented in the article. In particular, expansion of secant numbers for the sum of products grouped by the number of tangent numbers' cofactors with specification of corresponding coefficients in the expansion, representation (automatic expression) of elements of a sequence of alternating secant and tangent number through the previous ones, as well as a number of other correlations for the sequences and particular elements is determined.

Keywords: hyperbolic cosine distribution; cumulants; moments; numerical prism; cross-sections; secant numbers; tangent numbers.

Introduction

We obtained the numerical set {U(n;k,7)} structured as a «numerical prism» in the paper [1]. The construction of the numerical prism is based on relations between the cumulants xn and the initial moments an , where n e N, of the hyperbolic-cosine-type probability distribution defined [2, 3] by the characteristic function:

( b b Vm

f (t) = ch—t-imsh—t , where ¡j.,p,m e R ; m > 0, p * 0, i = \/-1. (1)

I m — m )

The obtained three-parameter distribution is described in [4] and is a generalization of the Meixner two-parameter distribution [5].

Theorem 1 ([1]). For a hyperbolic-cosine-type distribution with the characteristic function (1), the formula for calculating the cumulants xn, n = 1,2,..., is the following:

(2)

where Pn (b) are polynomials of the form:

Pn(b) = tv(n;7)bJ, n = 1,2,..., (3)

7=0

Tokmachev M.S. Correlations between Elements and Sequences

in a Numerical Prism

and the coefficients V(n; j) of the polynomials are calculated by the recurrence relation:

V(1; 1) = 1, V(1; j) = 0 for all j * 1, V(n +1; j) = ( j - 1)V(n; j -1) + ( j + 1)V(n; j +1) for n î N . (4)

The polynomials Pn (b) follow differentiation of the tangent function, and are known [6, 7] as Derivative polynomials.

Theorem 2 ([1]). For a hyperbolic-cosine-type distribution with the characteristic function (1), the formula for calculating the n-th order moments is the following:

M(Xn) =an =1 L Ь [ЩЕ\ = [L I XIU (n; k, j ) mk ^ , n = 0,1,2,..., (5)

( Y И

m j \ 0j y mj k=1 J=o {P,

where the moment-forming polynomials of two arguments Pn (m, b), n = 0,1,2,..., can be represented in the form:

Pn(m,b) = f fU(n;k, j)mkbJ , (6)

k=1 j=0

b ° / 0 , {U(n;k, j)} is a system of the integer coefficients of the polynomials related by the equations:

U(0;0,0) = 1, U(0;k, j) = 0 for any k, j * 0, U(n +1; k, j) = U(n;k -1, j -1) + (j - 1)U(n;k, j -1) + (j + 1)U(n; k, j +1), (7)

for n = 0,1,2,....

The identity U(n; 1, j) ° V(n; j) follows directly from equations (4) and (7). Therefore, the numbers V (n; j) are only special cases of numbers U (n; k, j) for k = 1.

For each fixed k , where k e N, the corresponding subset of {U(n;k, j)} forms a numerical triangle. In terms of geometry, the set {U(n;k, j)} is a set of numerical triangles ordered by k , and can be considered as a «numerical prism», where for each given k the triangle is a section of the prism [1, 8]. Similarly, we can obtain sections of the prism by fixing the argument j, j = 0,1,2,.... These sets are usually novel.

Let us either fix two of the parameters n, k, j, or express two of them through the remaining parameter. The numerical sequences obtained in the numerical prism include many well-known sequences (tangent numbers, secant numbers, Stirling numbers of the first kind, coefficients in expansions of various functions, for example, in the Bessel polynomials, etc.) [9]. However, the overwhelming majority of the obtained sequences is not mentioned in the literature. The issue of novelty is solved with the help of a sufficiently complete on-line encyclopedia of integer sequences (OEIS) [10].

We notice the lasting interest in numerical sequences and triangles. For example, the paper [11] presents a numerical pyramid which sections are known numerical sets: those of Deleham, Euler, MacMa-hon, Stirling. Also, the paper [11] gives a series of numerical sequences, among which only the sequences ,4000182, ,4000184, ,400147, and ^085734 in OEIS are common with the numerical prism. Other sequences and triangles are also present as subsets in the considered numerical prism. The paper [12] identifies the first elements of the numerical triangle {U(n;1, j)} as the numbers Tnk calculated

by the recurrence relation Tnk = S1k, Tn+1 k = (k - 1)Tn k-1 + (k + 1)Tn k+j. This is a special case of relation

(7). Along with interest in finding new numerical sequences, there is an increasing interest in classical sets of numbers considered from different points of view. In particular, the paper [13] presents simple and asymptotically fast algorithms for calculating tangent and secant numbers. In the considered numerical prism, we can not only easily calculate these sequences, but also establish structural relations between the sequences.

Cumulants and moments in the numerical prism structure

Therefore, the considered numerical prism is an ordered set {U (n; k, j)} of the coefficients of the polynomials Pn (b) and Pn (m, b) that form, respectively, the cumulants and moments of the hyperbolic-cosine-type probability distribution [1]. Moreover, according to (2) and in view of the equality

n

U(n; 1, j) ° V(n; j), the distribution cumulants are completely determined by the distribution parameters ¡¡,P,m and the set of coefficients {U(n;1, j)} :

Xn = mfl)"Pn f¡z 1 = m(lï £ V(n; j)fZ1J = m f4Y ¿U(n;1, j)bj , where b = Z . (8)

V P ) Vm ) ,-=o V P ) Vm ) /=0 P

m

For n = 1,12, j = 0,10, the corresponding section {U(n;1, j)} is partially presented in table 1. According to (7), subsequent values are calculated simply, without using differential relations. Note that the section {U(n;1, j)} is the known [10: ^008293] triangle of coefficients in the expansion of tg(n) x in powers of tgx .

Table 1

Section {U(n;1, j)} of the numerical prism

\ j n\ 0 1 2 3 4 5 6 7 8 9 10

1 1

2 1 1

3 2 2

4 2 8 6

5 16 40 24

6 16 136 240 120

7 272 1232 1680 720

8 272 3968 12096 13440 5040

9 7936 56320 129024 120960 40320

10 7936 176896 814080 1491840 1209600 362880

11 353792 3610112 12207360 18627840 13305600

12 353792 11184128 71867136 191431680 250145280 159667200

In order to calculate the initial moments an of the hyperbolic-cosine-type distribution (see (5)), we use the parameters p,p,m along with all sections of the numerical prism {U(n;k,7)} determined by k, where k = 1, n .

Let us consider the relation between the moments and cumulants with reference to the set

{U(n;k, 7)} .

As is well known, the moments {an} and cumulants {xs} of probability distributions are connected by the following polynomial relations:

a1 = X1; , 2

a2 = X2 + X1 ; a3 = X3 + 3 X1X2 + X13; a4 = X4 + (3X2 + 4X1X3) + 6Xi X2 + X14; (9)

«5 = C + (5CC4 + 1OC2C3) + (10CC3 + 15C1C22) + 10C13C2 +c5; «6 =C6 + (6C1C5 + 15^2^4 + 10C32) + (15C12C4 + 60C1C2C3 + 15C23) +(20C13C3 + 45c2C22)+15C14C2 +c6;

In the above expressions, the terms are grouped according to the number of cumulants-factors in the product. For any n, n e N , the relation takes the form:

= I I

k=1 +■■■+<?£ =n,

S, #0

aSi,S2,..;Sk Xsi X S2 ■■■Xsk

(10)

a

n

where k is the number of the multiplied cumulants, aare the corresponding coefficients at the products xs Xs2 ..Xsk, and si is the order of the cumulant xSj.

Theorem 3. For the hyperbolic-cosine-type distribution, consider the decomposition of the initial moments an into sum (10) of the cumulant products with given coefficients. Then the number of cumu-lants-factors in the corresponding terms is determined by the parameter k in the formula:

n n

= [ l \ Im ZU(n;k, j)bj .

m

k=1 j=0

Proof. As it follows from (5), (10), ß

= ["I IIU(n;к,j)mkb = I I I m ! 7Z7Z TT ,„ ,

к=1 j=0

к=1 ^1+ S2 +...+Sk=n, s, ¿0

si,s2,...,s^s^S2 ••• ASk

According to (8), the cumulants considered in (12) can be represented as:

Xst = m{£ 1" (s,;1, j)b.

m

(11)

(12)

(13)

j=0

Note that, regardless of the order s, of the cumulant, there is a factor m in equation (13). Consequently, there is a factor mk in each product of k cumulants, despite of different orders Sj,s2,...,sk of the cumulants xs1 Xs2 -Xsk .

Substitute the cumulants in the form of (13) into the right side of (12):

= [в I IIU(n;k,j)mkbj = I I

П

i.e.

a

= [l I IImk Iu (n; k, j)bj = [ в\ I~k

m

k=1 j=0

s1,s2,...,sk k=1 S1+ S2+...+Sk=n, i=1

s, ¿0

в \i Si

e\ m Iu (s, ;1, j)b m ! j=0

m

k=1 j=0

m

m

I

OL

П

s1,s2,...,sk

k=1 S1+ S2 +...+Sk=n, i=1

s, ¿0

IU (Si ;1, j)bj

j=0

(14)

where each term of the polynomials of variables m and b on the right side of (14) is such that the parameter k of mk indicates the number of cumulants-factors in the corresponding product. Since the polynomials on both sides of (14) are equal, coefficients at the same powers of m are also equal. Then, the following equality holds for any m:

m

k\ß_ m

Iu (n; ^ j )bj = [ mm I mk ^ .S1,S2,...,sk

j=0 V'"/ s1+s2 +...+sk =n, i=1

s, ¿0

I

П

IU (s, ;1, j)bj j=0

(15)

Therefore,

m

в IIU (n; k, j )bj = I

j=0

Os1,s2,...,skXs1 Xs2...Xsk for k = 1,2,...,n .

s1+s2 +...+sk =n, s, ¿0

At the same time, the power k and number of cumulants in the product are equal for each term on the right side of the equation.

This completes the proof of Theorem 3.

Theorem 4. There are the following relations for coefficients of the polynomials Pn (m,b):

IU (n; k, j)b = I Os1,s2,...,sk П

j=0 s1+s2 +...+sk =n, i=1

s, ¿0

IU (s, ;1, j)bj

j=0

where k = 1, n.

(16)

Proof. Divide both sides of (15) by (Pim) and compare expressions of the same powers of m in the polynomials.

а

n

n

а

n

s

s

s

Corollary 4.1. For any b, the following condition holds for the section {U(n;2, j)} of the numerical prism:

£ U (n;2,j)bJ = X

4s,l

J=0

s+l=n, s,l #0

£ U (s;1, js )b

Js

Js=0

£ U (l;l, j )bjl

Jl=o

In particular, compare coefficients of the same powers of b in the polynomials and obtain:

U (n;2, j) = £

V

s+l=n, s,l #0

£ U (s;1, js ) • U (l;1, j )

js+jl=j

, where j = 0, n .

(17)

(18)

Corollary 4.2. For any b, the following condition holds for the section {U(n;3, j)} of the numerical prism:

£ U (n;3, j)bj = £ a.

s,l

j=0

i+s+l=n, i,s,l #0

£ U (i;1, ji )bJi j=0

£ U(s;1, js )b

js

Js =0

£ U(l;1, j)bJl

Jl=0

(19)

In particular, compare coefficients of the same powers of b in the polynomials and obtain:

U (n;3, j) = £ au

s,l

i+s+l=n, i,s,l#0

£ U (i;1, j) •U (s;1, js ) • U (l;1, j )

ji + Js + jl = j

, where j = 0, n . (20)

Corollary 4.3. If b = 1, then relation (16) implies the following relation for coefficients of the polynomials Pn (m,b) :

n k

£ U (n;k, j) = £ as1,s2,...,sk П

j=0 s1+s2 +...+5k =n, i=1

s. #0

£ U(s. ;1, Ji )

Ji =0

, where k = 1, n.

(21)

Corollary 4.4. If b = 0, then relation (16) implyies the following relation for coefficients of the polynomials Pn (m, b) :

U (n; k ,0) = £ as1,s2,...,sk nU(s. ;1,0), where k = 1,

51+ s2 +...+sk=n, i=1

s. #0

In particular,

(22)

U(n; 1,0) = £ as1 U(s^1,0) = U(n;1,0) ;

s1=n,

U(n;2,0) = £ as,i U(s;1,0)U(l;1,0);

s+l=n, s,l

U(n; 3,0) = £ ahsJ U(i; 1,0) • U(s; 1,0) • U(l; 1,0).

i+s+l=n, i,s,l

Corollary 4.5. If b = 0, then the following relation is obtained from (22) by summation over k : £U(n; k, 0) = £ £ as1,s2,...,sk nU(s, ;1,0).

(23)

(24)

(25)

k=1

k =1 s1+ s2 +...+sk =n, s. #0

i=1

Theorem 5. In a section of the numerical prism{U(n;k,7)} , the numerical triangle {U(2n;k,0)} is a decomposition of the secant numbers {En} into the sum of products grouped by the number of factors of the tangent numbers {Tn}. Namely,

s

n

E0 = 1, En = Z Z a2s1 ,2s2,...,2skTs1 • Ts2 • ... • Tsk , (26)

k=1 2s1 +2s2+...+2sk=2n, S, *0

where n e N, a2s 2 2 are the positive integer coefficients in the representation, see (9), of the initial probability moments a2n = En as the sum of products of the cumulants X1,X2,...,X2n (2sj,2s2,...,2sk are even orders of the multiplied cumulants).

Proof. According to [8], if the parameters of the hyperbolic-cosine-type distribution are [ = 0, £ = 1, m = 1, then the cumulants {x2n} are tangent numbers x2n = Tn , n e N, and the initial moments {a2n} are secant numbers: E0 = 1, a2n = En, for n e N. In addition, for m = 0, p = 1, m = 1, relation (13) implies that:

X2„ = U(2n; 1, 0), x2„-1 = U(2n -1;1,0) = 0; (27)

and relation (11) implies that:

2n 2n-1

K2n ~

a2n = Z U(2n; k, 0), a2„-1 = Z U(2n -1; k,0) = 0, k=1 k=1 since U(2n -1; k, 0) = 0 for n, k e N .

We use relation (25) only for even n and st (since U(n; k, 0) and U(st ;1,0) are equal to zero, if n

and Sj are odd). Redesignate the indices and obtain:

2n 2n k

ZU (2n; k ,0) = Z Z «2s1,2s2,...,2sk nU (2s j ;1,0).

k=1 k=1 2s1 +2s2+...+2sk=2n, j=1

Sj

For nonzero summation indices, if 2sj + 2s2 +... + 2sk = 2n, i.e. Sj + s2 +... + sk = n , then k < n . Consequently, in the equality above, the summation over k is conducted from 1 to n. Therefore,

ZU(2n; k ,0) = Z Z «2s1,2s2,...,2sk №(2st ;1,0),

k=1 k=1 2s1 +2s2+...+2sk=2n, j=1

s,

i. e.

nk

a2n = Z Z a2s1,2s2,...,2sk n*2*,- • (28)

k=1 2s1 +2s2+...+2sk=2n, j=1

s,

Replace the moments and cumulants by the corresponding secant and tangent numbers a2n = En , X2s = Ts, in (28). We arrive at relation (26):

n

En = Z Z a2s1,2s2,...,2skTs1 ' Ts2 ' ...' Tsk .

k=1 2s1 +2s2+...+2sk=2n,

s,

This completes the proof of Theorem 5.

Theorem 6. For fixed k, k = 1,2,...,2n , each of sequences {U(2n;k,0)} is a sequence of such terms in the decomposition of secant numbers {En} that containk factors consisting of tangent numbers. Namely,

U(2n;k,0) = Z a2s1,2s2,...,2skTs1 'Ts2 '...'Tk , (29)

2s1 +2s2+...+2sk =2n, s,

En = ZU (2n; k ,0), (30)

n

k=1

where {Tt} are tangent numbers, a2 2 2 are positive integer coefficients in the representation, see (9), of the initial moments a2n = En as the sum of products of the cumulants Ci,C2, . ,C2« .

Proof. In order to obtain equation (29), replace n by 2n in (22) and use relation (27), where x2si = T . Relation (30) follows from (26) and (29).

Corollary 6.1. The sequence {U(2n; 1,0)} is a sequence of the tangent numbers {Tn} .

The statement directly follows from (29), since 2sx + 2s2 +... + 2sk = 2n, a2 2 2sk °a2n for k = 1, and all coefficients a2n of the highest cumulants %2n in formulas (9) are equal to 1.

Corollary 6.2. The sequence {U(2n;2,0)} satisfies the following relation:

U(2n;2,0) = £ a2s^ U(2s;1,0)U(2l;1,0) = £ a^ Ts • Tl .

2s+2l=2n, 2s+2l=2n,

s,l*0 s,l*0

The statement is a special case of (29) and (23), since Ts = x2s = U(2s; 1,0) for any s, se N.

Corollary 6.3. The sequence {U(2n;3,0)} satisfies the following relation:

U(2n;3,0) = £ a2i,2s,2l U(2i;1,0) • U(2s;1,0) • U(2l;1,0) = £ a^i T • Ts • Tl .

2i+2s+2l=2n, 2i+2s+2l=2n,

i,s,lï0 i,s,lï0

The statement is a special case of (29) and (24), since Ts = x2s = U(2s; 1,0) for any s, se N.

Using the formulas for representation of cumulants in terms of moments, we can also obtain inverse relations expressing the tangent numbers {Ti} in terms of the secant numbers {En} indicated in the section {U(n;k,0)} .

By analogy with Theorems 5 and 6, we can also represent relations between other sequences. Let us give two examples. Let m = 1, b = 1, m = 1. Then cumulants and moments of distribution connected by mutually inverse relations are the sequence (1, 2, 4, 16, 80, 512, 3904, 34816, ...) [10: ,4000831] and the sequence (1, 1, 3, 11, 57, 361, 2763, 24611, ...) [10: 4001586], respectively. The latter sequence is given by generalized Euler numbers. For m = 1, b = 1, m = 2, a sequence of cumulants, i.e. a sequence of alternating secant and tangent numbers, and a sequence of moments are equal up to a shift on two positions: xk = ak-2, k = 2,3,... [8]. Therefore, this sequence can be restored by the previous elements (autoexpression). If we know the structure (representation of cumulants through moments and, conversely, moments through cumulants), then we can use the numerical prism to find specific terms in the expression of each sequence through its pair sequence.

Remark 1. For 2n < 12, if all moments and cumulants of odd order are equal to zero, then the coefficients a2 2 2 used in the formulas to connect moments and cumulants of even order are as follows:

a2 = X2 ; a=i4+(3^22);

«6 =J6+(15^2^4)+(15J2); (31)

«8 = 18 + (28^2^6 + 35J42) + (210^4) + (105J24) ; «10 = *10 + (45^2^8 + 210^6) + (630^2^6 +1575J2J42) + (3150^) + (945j5) ;

«12 = X12 + (66Z2Z10 + 462J62 + 495^4^8) + (1485^8 +13860^4^6 + 5775J3) +

+(13860J23J6 + 51975^22^42) + (51975J24J4) + (10395J26) ;

Remark 2. In order to illustrate the theory, the fragment of section {U(n;k,0)} of the numerical

prism is given in table 2 (initial values for n = 1,16, k = 1,8 ). The rows present terms in the expansion of secant numbers by tangent numbers.

Conclusion

In the present work, the structured values of the numerical prism result in different relations and regularities in the field of numbers. At the same time, the numerical set {U (n; k, j)} has other applica-

tions. For example, for given coefficients {U (n; k, j)} and formulas for the density of the hyperbolic-cosine-type distribution [14], we can use the relation between the initial moments and derivatives of the characteristic function in order to obtain analytically values of improper integrals for a certain class of functions. For the parameters P,m, the papers [15-17] present the general form of exact formulas for the integrals of expressions containing a beta function with complex-conjugate variable arguments. In particular, according to [16], values of the integrals of a combination of a power function, an exponential function, a hyperbolic secant or cosecant, and some polynomials with different parameters are obtained for m e N.

Table 2

Section {U(n;£,0)} of the numerical prism

и\ 1 2 3 4 5 6 7 8

1

2 1

3

4 2 3

5

6 16 30 15

7

8 272 588 420 105

9

10 7936 18960 16380 6300 945

11

12 353792 911328 893640 429660 103950 10395

13

14 22368256 61152000 65825760 36636600 11351340 1891890 135135

15

16 1903757312 5464904448 6327135360 3918554640 1427025600 310269960 37837800 2027025

Therefore, the integer set {U (n; k, j)} introduced from probabilistic considerations is also of independent interest for mathematical analysis, number theory, coding theory, etc. A number of relations of the numerical prism and more detailed initial fragments of sections are presented in the author's monograph [17].

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17. Tokmachev M.S. Veroyatnostnye raspredeleniya: ot kharakterizatsii k chislovym mnozhestvam (Probability distributions: from characterization to numerical sets). Saarbryukken, LAP LAMBERT Publ., 2018, 224 p. (in Russ.).

Received December 7, 2018

Bulletin of the South Ural State University Series "Mathematics. Mechanics. Physics" _2019, vol. 11, no. 1, pp. 24-33

УДК 519.24; 511.3 DOI: 10.14529/mmph190104

СООТНОШЕНИЯ ЭЛЕМЕНТОВ И ПОСЛЕДОВАТЕЛЬНОСТЕЙ В ЧИСЛОВОЙ ПРИЗМЕ

М.С. Токмачев

Новгородский государственный университет им. Ярослава Мудрого, г. Великий Новгород, Российская Федерация E-mail: [email protected]

Рассматривается числовая призма, ранее введенная автором как упорядоченное множество в связи с исследованием трехпараметрического вероятностного распределения типа гиперболического косинуса, являющегося обобщением известного двухпараметрического распределения Майкснера. В геометрической терминологии элементы числовой призмы - коэффициенты мо-ментообразующих полиномов для указанного распределения, которые получаются с помощью как дифференциальных, так и алгебраических рекуррентных соотношений.

Каждый из бесконечного количества элементов зависит от трех индексов, которые и определяют его местоположение в призме. При фиксировании одного или двух индексов получаются сечения призмы: числовые треугольники или числовые последовательности. Среди сечений имеются широко известные, например, числовой треугольник Стирлинга, числовой треугольник коэффициентов в полиномах Бесселя, последовательности тангенциальных и секансных чисел и др. Однако подавляющее большинство числовых множеств в сечениях призмы ранее в литературе не встречались.

Ввиду структуры и алгоритма построения, сечения числовой призмы оказываются связанными между собой не только общей формулой построения, но и определенными соотношениями. Как результат, в статье представлены формулы связи между различными группами элементов.

32 Bulletin of the South Ural State University

Ser. Mathematics. Mechanics. Physics, 2019, vol. 11, no. 1, pp. 24-33

В частности, найдены разложение секансных чисел на сумму произведений, сгруппированных по количеству сомножителей тангенциальных чисел с указанием соответствующих коэффициентов в разложении, представление (автовыражение) элементов последовательности чередующихся секансных и тангенциальных чисел через свои предыдущие, а также ряд других соотношений для последовательностей и отдельных элементов.

Ключевые слова: распределение типа гиперболического косинуса; кумулянты; моменты; числовая призма; сечения; секансные числа; тангенциальные числа.

Литература

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14. Токмачев, М.С. Прикладной аспект обобщенного распределения гиперболического косинуса / М.С. Токмачев // Вестник НовГУ. - 2005. - № 34. - С. 96-99.

15. Токмачев, М.С. Некоторые интегралы, связанные с распределением типа гиперболического косинуса / М.С. Токмачев // Математика в вузе и в школе: Труды XXIV Международной научно-методической конференции. - СПб.: Петербургский гос. ун-т путей сообщения. - 2012. -С. 185-186.

16. Токмачев, М.С. Вычисление интегралов от функций некоторого класса с вероятностной интерпретацией / М.С. Токмачев // Вестник НовГУ. Сер.: Физ-мат. Науки. - 2014. - № 80. -С. 42-46.

17. Токмачев, М.С. Вероятностные распределения: от характеризации к числовым множествам / М.С. Токмачев. - Саарбрюккен: LAP LAMBERT Academic Publ., 2018. - 224 с.

Поступила в редакцию 7 декабря 2018 г.