MULTIPLE EULER TYPE INTEGRAL REPRESENTATIONS FOR THE KAMP´E DE F´ERIET FUNCTIONS Текст научной статьи по специальности «Математика»

Chelyabinsk Physical and Mathematical Journal. 2023. Vol. 8, iss. 4. P. 553-567.

DOI: 10.47475/2500-0101-2023-8-4-553-567

MULTIPLE EULER TYPE INTEGRAL REPRESENTATIONS FOR THE KAMPE DE fERIET FUNCTIONS

T.G. Ergashev1", A. Hasanov2,3,b, T.K. Yuldashev4,c

1 National Research University “Tashkent Institute of Irrigation and Agricultural Mechanization Engineers", Tashkent, Uzbekistan

2Romanovskiy Institute of Mathematics, National Academy of Sciences, Tashkent, Uzbekistan

3 Ghent University, Ghent, Belgium

4 Tashkent State University of Economics, Tashkent, Uzbekistan

"[email protected], [email protected], [email protected]

By the aid of Appell, Humbert and Bessel functions, the integral representations for a Kampe de Feriet function are found. The validity of integral representations for a Kampe de Feriet function of general form are proved. Conditions, under which these representations are expressed in terms of products of two generalized hypergeometric functions are found. Examples, in which the integral representation of the Kampe de Feriet function containing Appell, Humbert or Bessel functions, are identified.

Keywords: Kampe de Feriet functions, multiple Euler type integral representations, generalized hypergeometric functions of second order, Bessel function, Appell functions, Humbert functions.

1. Introduction

A great interest in the theory of hypergeometric functions (that is, hypergeometric functions of one, two and more variables) is motivated essentially by the fact that solutions of many applied problems involving thermal conductivity and dynamics, electromagnetic oscillation and aerodynamics, quantum mechanics and potential theory are obtainable with the help of hypergeometric (higher and special or transcendent) functions [1-3]. Such kinds of functions are often referred as special functions of mathematical physics.

It is known that hypergeometric series F(a, b; c; z) is studied by Leonhard Euler. Appell has defined, in 1880, four series: F1, F2, F3, F4. All of these series are analogous to Gauss’ series F(a,b; c; z). P. Humbert has studied confluent hypergeometric series in two variables.

The four Appell series were unified and generalized by Kampe de Feriet [4]. He defined a general hypergeometric series in two variables. The notation introduced by Kampe de Feriet for his double hypergeometric series of superior order was subsequently abbreviated by Burchnall and Chaundy [5]. Srivastava and Panda [6] gave the definition of a more general double hypergeometric series (than the one defined by Kampe de

Ы : (bq); (ck); x y (at) : (pm); (yra); ,y

and announced some groups of conditions on the parameters und

Feriet) in a slightly modified notation Fp

l:m;n

see, Equation (14)) er which the Kampe

The research is supported by the Intercontinental Research Center “Analysis and PDE” (Ghent University, Belgium), grant G.0H94.18N and by the Special Research Fund of Methusalem programme (Ghent University), grant 01M01021.

554

T.G. Ergashev, A. Hasanov, T.K. Yuldashev

de Feriet series converges in a non-empty set. Interesting results in this direction have been obtained in the works [7-14].

Many special functions appear as solutions of differential equations or integrals of elementary functions (for instance, see, [15-19]). In the works [20; 21], some Kampe de Feriet functions F, Fand F^q.’2 , F^:?, FЩ F^il, F^.:o;;2, Fow are studied. Thanks to properties of Kampe de Feriet functions, authors manage to obtain a solution to one boundary value problem for the differential equation in explicit form.

Integral representations are very important in the study of applied problems. For evaluations and extensions of results on Euler type integrals, we refer a paper [22]. Also, in this regard, it is noticed that the general sextic equation can be solved in terms of Kampe de Feriet function (see, [23] and [24]). Therefore, well-known reference books [25-27] are highly respected among applied scientists, in which second-order hypergeometric functions in one and two variables are considered. Hasanov and Ruzhansky in [28] constructed Euler-type integral representations for 205 second order hypergeometric series in three variables. However, there are very few works on integral representations of hypergeometric functions when their order exceeds two. We note only work [29], in which 18 integral representations are constructed for some Kampe de Feriet functions of the fourth order.

In this paper we will obtain the multiple Euler type integral representations for the Kampe de Feriet functions of arbitrary order.

2. Preliminaries

We consider the familiar (Euler’s) Gamma function r(z), defined for z € C \ Z— by the following formula

✓ CO

f tz-1e-tdt (Re(z) > 0),

о

r(z)

< r(z + n)

n— 1

П(z + 3) j=о

(z € C \ Z—; n € N).

The general Pochhammer symbol (or the shifted factorial) (A)v (A, v € C) is defined by thr formula

(v — 0; A € C \ {0})

(v — n € N; A € C),

(A — —n; v — k; n,k € N0; 0 ^ k ^ n),

(A — —n; v — k; n, k € N0; k > n),

(v — —k; k € N; A € C \ Z),

where it is understood conventionally that (0)0 :— 1 and assumed tacitly that the Gamma quotient exists. Hence, the following standard notations are used:

N :— {1,2, 3,...} , N0 :— NU {0} and Z— :— Z— U {0} — {0, —1, —2, ...} .

Moreover, as usual, the symbols C, R, N, Z, R+, and R— denote the sets of complex numbers, real numbers, natural numbers, integers, positive, and negative real numbers,

(A)v :—

r(A + v) r(A)

1

n—1

П (A + 3)

j=о

(—1)k n!

(n — k)!

0

(—1)k , (1 — A)k

Multiple Euler type integral representations for the Kampe de Feriet functions

555

respectively. The Beta function is defined by the following integral

B(a, в) := [ ta-1(1 - t)e-1dt = } , Rea > 0, Re^ > 0.

J Г(а + в)

0

The celebrated Gauss hypergeometric function

a, b;

F(a, b; c; z) = F

c ;

E G# , c =0,-1,-2,...

k=0

(c)k k!

is contained in the generalized hypergeometric function pFq, involving p numerator parameters, a1, ...,ap, and q denominator parameters, and b1, ...,bq, as a special case. Following the standard notations and conventions, we define it here as follows [30,

p. 182]:

\ ^ П (aj )k

(ap) ;

F

pF q

(bq);

,Fq [(ap) ; (bq) ; z] : = 'У ]

те H K^JJk k

j=i zk

q k!

k=0 П (bj )kk! j=1

Gauss' series (2) in the present notation has the form

a, b;

2Fi(a,b; c; z) = F The familiar Bessel function

:= F(a, b; c; z).

Ja(z) = t

(-1)m

m=0

*)

т!Г(т + a + 1) \2J

z \ 2 m+a

also belongs to the class of generalized hypergeometric functions pFq:

)Fi(—; a; —z) = oFi

z

a;

r(a)z(1-a)/2Ja-i (2^^) .

The great success of the theory of hypergeometric series in one variable is that this theory has stimulated the development of a corresponding theory in two and more variables. Four Appell functions are defined as follows [31]:

Fi(a,b,b’; c; x,y) = (

m,n=0 '

(a)m+n(b)m (b )

m \w Ы^тп n

c)m+nm!n!

xy

(3)

F2(a,b,b';c,c';x, y) = X!

m,n=0

F3 (a,a',b,b’; c; x,y)= m

m,n=0

(a')m+n(b')m (b )n xmy n

(c)m (c')n m!n! ,

(a)m (a )n (b)m (b )n xmyn x y

c)m+nm!n!

F4 (a.b;cV;x,y) = ±

m,n=0 ( )m ( )n ! !

(6)

where, in all definitions (3)-(6), as usual, the denominator parameters c and d are neither zero nor a negative integer.

c

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T.G. Ergashev, A. Hasanov, T.K. Yuldashev

Seven confluent forms of the four Appell series were defined by Humbert in [32], and these confluent hypergeometric series in two variables are denoted by

ф1(a,e; y; x,y) = £ (y) + m!n! xmyn, |x| <1 m,n=0 ( ))m+n ! ! (7[

ф (Y^nminTy’- m,n=0 4 7 (8

(e) ф 03; Y; x,y)= £ ,,e )mm,n, xmyn, m,n=0 ())m+n m!n! (9

iTr ( 0 / \ \ л (a)m+n (в)m m n 1 1 -1 Ф1 (a,e; y,Y; x,y)= У, f wo 1 Iх y , |x| < 1 n (Y)m(7')nm!n! m,n=0 4 7 4 7 (10

^ ( ) *2 yY x-y)= £ (Y)m (-Win!xmy”’ m,n=0 n (11

H (a.a'.ft Y; x,y)= £ (Q()m (Q°" (f)m xmyn, |x| < 1, m,n=0 (Y)m+n m!n! (12

H2(a,e; y ; x,y)= £ ,,m I .^у’ |x| < 1, m^0 (Y)m+n m!n! (13

where the denominator parameters 7 and -' are neither zero, nor a negative integer. Hypergeometric functions, which defined in (7)-(13) are called Humbert functions.

Just the Gaussian series F(a, b; c; z) was generalized to pFq by increasing the numbers of the numerator and denominator parameters. The four Appell series were unified and generalized by Kampe de Feriet in [4] and defined a general hypergeometric series in two variables (see, [33, p. 150, eq.(29)]). The notation introduced by Kampe de Feriet for his double hypergeometric series of of superior order was subsequently abbreviated by Burchnall and Chaundy [5, p. 112]. Srivastava and Panda (see, [6] and also [34, Section 3.1]) gave the definition of the more general double hypergeometric series (than the one defined by Kampe de Feriet) in a slightly modified notation

FP'-q,k

l:m,n

Ы : (bq) ; (ck);x y

(ai): (em); (Yn); ,

Fp:q,k a1 , ..., ap : b1, ...,

l:m,n a1, ... al : ^1, ...,

p

(aj)r s

= £ j=1 l

r,s=0 П (ai)r+s

5 •••) uq > •••) ^k?

x,y

m

q

k

j=1

H \u])r 11 \4)s r s j= 1 j=1 x_y_

m n r! o! ’

П (в )r П (-j )s ! !

j=1 j=1

n

where p,q,k,l,m,n E N U {0}, and for convergence of series we put

(i) p + q < l + m + 1, p + k < l + n +1, |x| < ro, |y| < ro or

(ii) p + q = l + m + 1, p + k = l + n +1, and

VO--1’ + |y|

max {|x| ,

1/(p-1’ < 1

|y|} < 1,

if p > l, if p ^ l.

(14)

Although the double hypergeometric series defined by (14) reduces to Kampe de Feriet series in the special case: q = k and m = n, yet it is usually referred in the literature as the Kampe de Feriet series.

Multiple Euler type integral representations for the Kampe de Feriet functions

557

3. Integral Representations

Theorem 1. Let p, q, k, l, m, n, K, Q E N U {0} and Aj, Bj E C, j E N. If

Re (Aj) > 0, Re (Bj) > 0, j E N,

then for every K and Q the following integral representation formula

:i5)

F

p : q+N—Q ;k+N—K

l+N: m

X

v fp: q ; k

A F l :m+Q;n+K

(ap) : (bq) , (AN—Q) ; (ck) , (BN—K) ; x y

(ai), (an + bn): (em) ; (Yn) ;

1 1 N

=cnJ .J n ft’—1 (i - tj )Bj

^0 0 y j=1

N times

(ap) : (bq) ; (ck) ;x у

(al) : (em) , (AN—Q+1,N) ; (Yn) , (BN—K+1,N) ; ,

dTN, (16)

is valid, where

(AN + BN) := (A1 + B1, ■■■■, AN + BN) ;

(An—q+1,n) := (An—q+1,-,An) , if Q E N; (An—q+1,n) := 0, if Q = 0; (Bn—k+1,n) := (Bn—k+1,-,Bn) , if K E N; (Bn—k+1,n) := 0, if K = 0;

N г

C,

N

N

N

. X := x tj, у := y (1 - tj) , dTN := dt,...df

N ■

j=1

j=1

Proof. The equality (16) follows easily from the definition of the Kampe de Feriet series (14), if we use the formula (1) for calculating the Beta function. □

Corollary 1. Let the conditions (15) be satified. If p = 0 and l = 0, then the Kampe de Feriet function defined in (14) can be represented as an integral of the product of two generalized hypergeometric functions

F 0:q+N—Q;k+N—K FN: m ; n

X

X q F

q Fm+Q

- :(bq) , (AN—Q) ; (ck) , (BN—K);

(AN + BN) : (em) ; (Yn) ;

1 1 N

= CN / ■ J П 1 (1 - tj)

0 0 y j=1 Ntimes

(bq) ;x

(em) , (AN—Q+1,N) ;

k ТП+к

(ck )

(Yn), (bn—k+1,n);

у

dTN ■ (17)

n

We consider some interesting examples in the concrete cases of parameters of the functions:

F1:2;1

F 1:1;0

a : b,A1; Bp

A1 + B1: e ;— ;

x,y

cj tAl—1 (1 — t)Bl—1 Fjig

— ::(в;;—;xt,y(1 — t)

dt, (18)

1

0

558

T.G. Ergashev, A. Hasanov, T.K. Yuldashev

F1:2;1 F 1:1;0

a : bi,b2]Bi,

Ai + Bi: e ;— ;

x,y

= Ci tAl-i (1 - t)Bl-i F 1:2;0

a : b1 , b2; ; , / -t ,\

- :e.A!;-;Xt'y(1 - t)

dt,

F 1:4;4 F 4:0;0

a : b, (A3);b', (B3); xy

C, (A3 + B3) : — ; — ;

1 1 1 о

C3

П V-1 (1 - tj)Bj-1 Fi (a, b, b; c; X, Y) dT:i,

000

j=i

F1:4;4 F 3:1;1

a : ^ (A3);b', (B3);xy

(A3 + B3): c ; c/ ; ’

i i i „

C3 П j-1 (1 - tj)B-^ F2 (a,b,b/; c,c/; X,Y) dT3

000

j=i

F1:3;3

F3:0;0

a : b, Ai, A2; b/, Bi, B2;

(A3 + B3): - ; - ;X’y

1 1 1 о

C3 / Д j-1 (1 - tj )B-1 F2 (a,b,b/; A3,B3; X,Y) dT3

000

j=1

F1:4;3

F3:1;0

a :b, (A3); b/,Bi,B2; xy

(A3 + B3): c ; - ; ’

1 1 1

= CJ П j-1 (1 - tj)B-^ F2 (a, b, b/; c,B3; X,Y) dT3

000

j=1

F1:3;4

F3:0;1

a : b,Ai,A2;b/, (B3);xy

(A3 + B3): - ; c ; ’

1 1 1 о

C3/ / / П j-1 (1 - tj)B-1 F2 (a, b, b/; A3, c; X,Y) dT3

000

j=1

f0:5;5

F 4:0;0

:a,b, (Aз);a/,b/, (B3);xy c, (A3 + B3) : - ; - ;

1 1 1 о

= С3П / П j-1 (1 - tj)B-1 F3 (a, a/, b, b/; c; X, Y) dT3

j=1

1

(19)

(20)

(21)

(22)

(23)

(24)

(25)

000

Multiple Euler type integral representations for the Kampe de Feriet functions

559

F 3:1;1

а,Ъ :{Аз)-,{Вз)-,т y

(Аз + Вз) : c ; d ; ’

1 1 1

= Сз / Д j-1 (1 - tj)Bj-1 F (а, Ъ; c, c'; X,Y) dT:i,

0 0 0

j=1

2:2;3

F з:0;1

а, Ъ :A1, A2; (Вз);

(Аз + Вз): -; с;1'»

1 1 1 о

Сз

П j 1 (1 - tj )B-1 F (а, Ъ; Аз,c; X,Y) dTз,

000

j=1

2:з;2 F з:1;0

а, Ъ :(Аз); В1,В2;

(Аз + Вз) : с; - ;*■»

1 1 1 О

Сз

П j 1 (1 - tj)B-1 F (а, Ъ; с, Вз; X,Y) dT3,

2:2;2 F з:0;0

0 0 j=1

В1 В2;

;x,y -;

1 1 з

Сз

П j 1 (1 - tj)Bj-1 F4 (а, Ъ; Аз, Вз; X,Y) dT3,

000

j=1

F1:4;з F 4:0;0

а :Ъ, (Аз);(Вз);т»

d (Аз + Вз) : - ; - ;

1 1 1

= Сз М / П j'-1 (! - tj)B"‘1 Ф1 (а, Ъ; с;X,Y) dT>,

000

j=1

F0:4;4 F 4:0;0

- : Ъ, (Аз); Ъ', (Вз); ‘

c, (Аз + Вз) : - ; - ;

1 1 1 о

= Сз / / / П -1 (1 - tj)B'-1 Ф2 (Ъ, Ъ'; c; X,Y) dft,

000

j=1

F0:4;з

F 4:0;0

:Ъ, (Аз);(Вз); c, (Аз + Вз) : - ; - ;

1 1 1

= Сз/ / / П j-1 (1 - tj )B'-1 Фз (Ъ; c; X,Y) dT>,

j=1

(26)

(27)

(28)

(29)

(30)

(31)

(32)

000

560

T.G. Ergashev, A. Hasanov, T.K. Yuldashev

F1:4;3 F 3:1;1

a :b, {Аз)-,{Вз)-,т y (Аз + Вз) : c ; d ; ’

1 1 1 о

= Сз /П j-1 (1 - tj)B-^ Ф1 (a , b; c ,d; X ,Y) ^ ,

0 0 0

j=1

(33)

F1:з;2

F з:0;0

a : b,A1,A2; B1,B2;

(Аз + Вз): - ; - ;x’y

1 1 1

Сз

з

П V 1 (1 - tj)B-1 Ф1 (a, b; Аз,Вз; X,Y) dT^ (34)

000

j=1

F1:4;2 F з:1;0

a :b, (Аз);В1,В2;

(Аз + Вз): c ; - j

111.

x,y

Сз/ / / Д j-1 (1 - tj )Bj-1 Ф1 (a, b; с,Вз; X,Y) dT^ (35)

000

j=1

F1:з;з F з:0;1

a : b, А1, А2; (Вз) (Аз + Вз) : - ; c

x,y

1 1 1

Сз

П j 1 (1 - tj)B-1 Ф1 (a, b; Аз, c; X,Y) dT^ (36)

000

j=1

F1:з;з F з:1;1

a :(Аз);(Вз);

(Аз + Вз): c ; d ;Х,У

1 1 1

= Сз

П V 1 (1 - tj)Bj-1 Ф2 (a; c,c'; X,Y) dT^ (37)

000

j=1

F1:2;2 F з:0;0

a :АЪ А2;ВЬ В2;

(Аз + Вз): -; - ;*■»

1 1 1 о

Сз / / / Д j-1 (1 - tj)Bj-1 Ф2 (a; Аз,Вз; X,Y) dT^ (38)

000

j=1

F1:з;2 F з:1;0

a : (Аз) ; B1, В2;

(Аз + Вз): c; -;x’y

1 1 1

Сз

П j 1 (1 - tj)B-1 Ф2 (a; c, Вз; X,Y) dT^ (39)

j=1

000

Multiple Euler type integral representations for the Kampe de Feriet functions

561

F1:2;3 F 3:0;1

a :Al, A2; m ;

(A3 + B3) : — ; c ;

x,y

c ;

i i i

П j 1 (1 - tjФ2 (a; A3, c; X,Y) dT:i,

000

j=i

F0:5;4 F 4:0;0

:a,b, (A3); a', (B3);

c, (A3 + B3) : — ; — ;

1 1 1 о

x,y

= C3 / П j" (1 — tj)^7'"1 S1 (a, a', b; c; X,Y) dT:i,

000

j=1

0:5;3 F 4:0;0

:a,b, (A3);(B3);x y

(A3 + B3): — ; — ;x'y

111.

C3

000

П j" (1 — tj)B*-1J H2 (a, b; c; X, Y) dT>,

j=1

F 0:N ;N FN : 0 ; 0

:(an );(bn ); (an + bn) : a e ;

x—y

= C„(v^) 1-“ (yy)1-e x

1 1

X

N

П |tf*-41 (1—tj)B’- ^

0 0 ^j=1

N times

Ja-1 [2vXj Jfi-1

dTN,

F 1 :N +1;N+1 FN+1: 0 ; 0

a : b, (an);b', (bn);xy

c (an + bn) : — ; — ;

1 1

C

N

N

П V 1 (1 — tj )B*-1 F1 (a, b, b'; c; X,Y) dTN,

о 0 yj=1

N times

j-r 1: N+1;N +1 FN: 1 ; 1

a :b, (AN); ^ (BN);

(an + bn ) : c ; c' ; ’

1 1

C,

N

N

П tj* 1 (1 — tj)B* 1 F2 (a, b, b'; c, c'; X,Y) dT,,

о 0 ^ j=1

N times

F 1:N ;N

FN: 0 ; 0

a : b, (AN-1) ; ^ (BN-1)

(A, + Bn) :

.x,y

1 1

N

CW ... Щ tf 1 (1 — tj)B*-1 F2 (a, b, b'; An,Bn; X,Y) dT,,

00

j=1

(40)

(41)

(42)

(43)

(44)

(45)

(46)

N times

562

T.G. Ergashev, A. Hasanov, T.K. Yuldashev

rp 1:N+1;N FN: 1 ; 0

a : b, (An); b', (Bn-i) ;

(An + Bn ): c ; - ;Х,У

i i

= C

N

N

П v-1 (! -jA-1

N times

F (a, b, b'; c, Bn; X, Y) dTN,

(47)

T711:N ;N +1 FN:0; 1

a : b, (An-1) ; b', (Bn); (An + Bn) : — ; c ; ’

Ntimes

-1 (1 - j)B

F (a, b, b'; An, c; X, Y) dTN,

(48)

F 0 :N+2;N+2 FN+1: 0 ; 0

- :a,b, (An); a',b', (Bn);

c (An + Bn) : - ; - ; J

Ntimes

j-1 (1 - tj )B

F3 (a, a', b, b'; c; X, Y) dTN,

(49)

j-r 2:N ;N FN: 1; 1

a, b :(An);(Bn);

(An + Bn): c ; c' ;X,y

N times

j-1 (i - tj )Bj-1

F4 (a, b; c, c'; X, Y) dTN,

(50)

2:N-1;N-1 FN: 0 ; 0

a b : (An-1);(Bn-1); (An + Bn) : - ; - ; ’

N times

j-1 (1 - tj )B

F4 (a, b; An, Bn; X, Y) dTN,

(51)

F 2:N ;N-1

FN: 1; 0

a b :(An); (Bn-1); x y

(An + Bn ) : c ; - ; ’

A-1 (1 - tj )B

F4 (a,b; c, Bn; X, Y) dTN, (52)

N times

Multiple Euler type integral representations for the Kampe de Feriet functions

563

p2 :N-1;N FN: 0 ; 1

a, b : (An-i) ; V, (Bn);

(An + Bn) : — ; c ,

i i

.x,y

N

Cn ... ПН? 1 (1 - tj)Bj-1 F (a, b; An , c; X,Y) dTN,

о о yj=1

N times

F-,

1 :N+1;N

N+1: 0 ; 0

a : b, (An); (Bn);

c (An + Bn ): — ; — ; ’

1 1 N

Cn ... П№-1 (1 - tj )Bj-1 Ф1 (a, b; c; X,Y) dTN,

0 0 ^ j=1 Ntimes

F 0 :N +1;N+1 FN+1: 0 ; 0

- : v (An); b/, (Bn); x y

c, (An + Bn) : — ; — ; ’

1 1

N

Cn ... /П t? -1 (1 - tj)Bj-1 Ф2 (b, V; c; X,Y) dTN

0 0 y j_1 Ntimes

F 0 :N+1;N FN+1: 0 ; 0

- : Ь, (An); (Bn) ; x y

c (An + Bn): — ; — ;

1 1

N

N

П V-1 (1 - tj)Bj-1 Фз (b; c; X, Y) dTN,

0 0 y j=1 Ntimes

1:N+1;N FN: 1 ; 1

a :b, (An);(Bn);xy

(An + Bn) : c ; c ; ’

1 1

N

Cn / ... Щ t? -1 (1 - tj)Bj-1 Ф1 (a, b; c,c'; X,Y) dTN

0 0 ^ j=1 N times

F 1:N ;N-1

FN: 0; 0

a : b, (AN-1) ; (BN-1) ;

(An + Bn ) : - ; - ; ’

1 1

N

Cn/ ... Щ t? 1 (1 - tj)Bj-1 Ф1 (a, b; An,Bn; X,Y) dTN,

00

j=1

(53)

(54)

(55)

(56)

(57)

(58)

Ntimes

564

T.G. Ergashev, A. Hasanov, T.K. Yuldashev

rp 1:N+1;N —1 FN: 1 ; 0

a :b, (An) ;(Bn—1);

(An + Bn ): c ; - ;Х,У

1 1

= C,

N

N

П1/—1 (1 - tj )Bj—1

N times

Ф1 (a, b; c, Bn; X,Y) dTw,

(59)

j-r 1:N ;N FN: 0 ; 1

a : b, (AN—1);(BN);

(An + Bn) : — ; c ; ’

Ntimes

jj — 1 (1 - tj)B

Ф1 (a, b; An , с; X, Y) dTN,

(60)

1:N;N fN : 1; 1

a :(An );(Bn ); xy

(An + Bn ) : c ; c/ ; ’

Ntimes

— 1 (1 - tj)B

Ф-2 (a; c, с/; X, Y) dTN,

(61)

1:N—1;N— 1 fN: 0 ; 0

a :(An—1);(Bn —1) ; x y

(An + Bn ): - ; - ;

N times

tAj 1 (1 - tj)Bj—1

Ф2(a; An , Bn ;X, Y) dTN,

(62)

7-11:N ;N —1 fN : 1; 0

a :(An );(Bn—1);ху

(An + Bn ) : c ; - ; ’

N times

;fj — 1 (1 - tj)B

Ф2(a;c, Bn ;X, Y) dTN,

(63)

F 1:N—1;N fN: 0 ; 1

a : (An—1);(Bn );ху

(An + Bn ) : - ; c ; ’

^Aj—1 (1 - tj)B

Ф2 (a; An , с; X, Y) dTN,

(64)

N times

Multiple Euler type integral representations for the Kampe de Feriet functions

565

F 0 :N+2;N +1 FN+1: 0 ; 0

- :a,b, {An);a', {Bn);

C {An + Bn) : — ; — ; ’

i i

N

N

П j-1 - j)B

о о yj=1

N times

1 (a,a',b; c; X,Y) dTN, (65)

F 0 :N+2;N fN+1: 0 ; 0

- : a,b, {An); {Bn); x y

c, {An + Bn) : — ; — ; J

Ntimes

tAj 1 (1 — tj )Bj-1

'2 (a, b; c; X,Y) dTy. (66)

Integral representations (17)-(66) are easy to prove using the definition (1) of the Beta function.

References

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13. Choi J., Rathie A.K. General summation formulas for the Kampe de Feriet function. Montes Taurus Journal of Pure and Applied Mathematics, 2019, vol. 1, iss. 1, pp. 107-128.

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14. Choi J.J., Milovanovic C.V., Rathie A.K. Generalized summation formulas for the Kampe de Feriet function. Axioms, 2021, vol. 19, no. 4, p. 318.

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21. Bin-Saad M.G., Ergashev T.G., Ergasheva D.A., Hasanov A. The confluent Kampe de Feriet series and their application to the solving of the Cauchy problem for degenerate hyperbolic equation of the second kind with the spectral parameter. Mathematica Pannonica. New Series, 2023, no. 1, pp. 1-15.

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567

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Article received 08.08.2023.

Corrections received 21.09.2023.

Челябинский физико-математический журнал. 2023. Т. 8, вып. 4- С. 553-567.

УДК 517.552 DOI: 10.47475/2500-0101-2023-8-4-553-567

КРАТНЫЕ ИНТЕГРАЛЬНЫЕ ПРЕДСТАВЛЕНИЯ ТИПА ЭЙЛЕРА ДЛЯ ФУНКЦИЙ KAMPE DE FERIET

Т. Г. Эргашев1", А. Хасанов2,3 6, Т. К. Юлдашев4,с

1 Национальный исследовательский университет «Ташкентский институт инженеров ирригации и механизации сельского хозяйства»», Ташкент, Узбекистан 2Институт математики им. В. И. Романовского АН Узбекистана,

Ташкент, Узбекистан

3 Университет Гента, Гент, Бельгия

4 Ташкентский государственный экономический университет, Ташкент, Узбекистан "[email protected], [email protected], [email protected]

C помощью функций Аппеля, Гумберта и Бесселя найдены интегральные представления для функции Kampe de Feriet. Доказана справедливость интегральных представлений для функции Kampe de Feriet общего вида. Найдены условия, при которых эти представления выражаются через произведения двух обобщённых гипергеометрических функций. Приведены примеры, в которых интегральное представление функции Kampe de Feriet содержит функции Аппеля, Гумберта или Бесселя.

Ключевые слова: функции Kampe de Feriet, кратные интегральные представления типа Эйлера, обобщённые гипергеометрические функции второго порядка, функция Бесселя, функция Апелля, функция Гумберта.

Поступила в редакцию 08.08.2023.

После переработки 21.09.2023.

Сведения об авторах

Эргашев Тухтасин Гуламжанович, доктор физико-математических наук, доцент, профессор кафедры высшей математики, национальный исследовательский университет «Tашкентский институт инженеров ирригации и механизации сельского хозяйства», Ташкент, Узбекистан; e-mail: [email protected].

Хасанов Анвар, доктор физико-математических наук, профессор, главный научный сотрудник, Институт математики им. В. И. Романовского АН Узбекистана, Ташкент, Узбекистан; e-mail: [email protected].

Юлдашев Турсун Камалдинович, доктор физико-математических наук, доцент, профессор кафедры общих и точных дисциплин Ташкентского государственного экономического университета, Ташкент, Узбекистан; e-mail: [email protected].

Исследование поддержано Межконтинентальным исследовательским центром «Анализ и уравнения в частных производных» (Университет Гента, Бельгия), грант G.0H94.18N, и Специальным исследовательским фондом программы «Иерусалим» (Университет Гента), грант 01M01021.