CC BY
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Вестник КРАУНЦ. Физ.-мат. науки. 2018. № 1(21). C. 32-47. ISSN 2079-6641
DOI: 10.18454/2079-6641-2018-21-1-32-47
MSC 33C20, 35A08
SOME PROPERTIES OF HORN TYPE SECOND ORDER DOUBLE HYPERGEOMETRIC SERIES
Anvar Hasanov1, Maged G. Bin Saad2, Ainur Ryskan3
1 Institute of Mathematics, 81 Mirzo-Ulugbek Street, Tashkent 700170, Uzbekistan
2 Department of Mathematics, Aden University, Aden, Kohrmakssar, P.O.Box 6014,
Yemen
3 Institute of Mathematics, Physics and Computer science, KazNPU named after Abai,
86 Tole bi street, Almaty 0500012, Kazakhstan
E-mail: anvarhasanov@yahoo.com
Horn [1931, Hypergeometrische Funktionen zweier Veranderlichen, Math. Ann., 105(1), 381-407], (corrections in Borngasser [1933, Uber hypergeometrische funkionen zweier Veranderlichen, Dissertation, Darmstadt], defined and investigated ten second order hyperge-ometric series of two variables). In the course of further investigation of Horn's series, we noticed the existence of hypergeometric double series H| analogous to Horn's double series H2. The principal object of this paper is to present a natural further step toward the mathematical properties and presentations concerning the analogous hypergeometric double series H| Indeed, motivated by the important role of the Horn's functions in several diverse fields of physics and the contributions toward the unification and generalization of the hyper-geometric functions, we establish a system of partial differential equations, integral representations, expansions, analytic continuation, transformation formulas and generating relations. Also, we discuss the links for the various results, which are presented in this paper, with known results.
Key words: Gauss hypergeometric function, Horn double series, partial differential equations, integral representations, transformation, generating functions.
© Hasanov A., Saad M. G. B., Ryskan A., 2018
1. Introduction
The various interpretations of Gauss hypergeometric function
2F1 fc b;c;x) = £ m, (1.1)
m=0 (c)m m-
with parameters a, b, c e C and c </ Z ^ 0, where the Pochhammer symbol (a)m is defined by (a)m = r(a+m)/r(a) = a(a + 1)...(a+m — 1) for m ^ 1, (a)0 = 1 and r: Gamma function, have challenged mathematicians to generalize this function. The series (1.1) is easily seen to be convergent when |x| < 1. The great successes of hypergeometric series theory in one variable has stimulated the development of a corresponding theory in two and more variables (cf. [1], [2], [3], [5], [6], [7], [10], [11], [14]). The class of Horn's double Gaussian second order hypergeometric series consists of ten complete series (cf. [8], [9], [17]: G1,G2,G3,H1,...,H7. In the course of further investigation of Horn's series, we noticed the existence of double series analogous to the Horn's double series H2 of the form:
(a) 2m (b1)„(b2) n—m m_,n <->\
(C) m>n> ^ ^ . ( 2
m,n=0 (c)nm-Ai-
The series (1.2) is analogous to and modified form of the Horn's double series H2 [[17], p. 24(10)]:
H r b f 1 V (a)m—n(b)m(e)n(f)n xmyn (1 3)
H2 [a,b ,e,f; c; x,y] = £ -(c)-—. (1.3)
m n=0 ( )m
In this paper, we will study some of the properties of the series (1.2) involving differential equations, integral representations, expansions, generating functions and its relations with other known classical functions and hypergeometric series. We shall denote series (1.2) symbolically as
H (a, b1, b2; c;x,y) = £ ^«¿»y.. (1.4)
m,n=0 (c)nm-n-
According to the Horn's theory [13], it can be easily verified that the series (1.4) is the second order hypergeometric series. Clearly, in view of the formula
(A)2m = 22m(|)m(| + 2)m, m = 0, 1, 2,..., (1.5)
from (1.3) and (1.4) we obtain:
H (a,b1,b2;c;x,y) = H [b2,b1, f, f + 2;c;y,4x]. (1.6)
Analogous and symmetric functions from the set of different hypergeometric functions of two or more variables have attracted the greatest attention because these kinds of hypergeometric series have simple and elegant results.
2. System of partial differential equations
According to the theory of multiple hypergeometric functions (see [2]), the system of partial differential equations for Horn-type function is readily seen to be given as:
+- 0 ('+4)x-1 "(a+^+0 (a+^
c+yI) 0+y|y)y-1 -(b1 + 4) (b2+4-41 "=0
u = 0
(2.1)
where u = H\ (a,bi,b2;c;x,y). Now by making use of some elementary calculations, we find the following system of second-order partial differential equations:
x (1 + 4x) Uxx - yuxy - [b2 - 1 - 2 (2a + 3)x] Ux + a (a + 1) u = 0, ^ .)
y (1 - y) Uyy + xyUxy +[c - (bi + b2 + 1) y] Uy + bixUx - bib2U = 0. ( . )
It is noted that the two equations of the system (2.2) are linearly dependent, because the hypergeometric function satisfies the system. Now, in order to find the linearly independent solutions of the system (2.2), we consider u as in the form u = xTyvw an unknown functions, and t and v are constants which are to be determined. So, substituting u = xTyvw into the system (2.2), we obtain
y
x (1 + 4x) wxx - ywxy - [02 - 1 - 2t + v - (4a + 6 + 8t) x] wx - t-wy
x
+
T (T v + 4t (t - i) + «(a + 1) + (4a + 6) t
w = 0
x
y (1 - y) Wyy + xywxy + [Y + 2v - (2v - T + ßi + ß2 +1) y] Wy +(v + ßi) XWx
V (v + Y - 1)
(2.3)
+
y
+ (T - ß1 - ß2 - V) V + ß1T - ß1ß2
w = 0
It is noted that the system (2.3) is analogical to the system (2.2) and it is not difficult to see that the system satisfies the following solutions: for t = 0, v = 0, u1 = (a,0, y; S;x,y) and for t = 0, v = 1 - y, u2 = y1-YH| (a, 1 - y + ft, 1 - Y + 02; 2 - Y;x,y).
3. Integral representations
Integral representations for hypergeometric series are very useful. Change of variables in these integrals leads to equivalent integrals. This provides an effective and easy method to derive certain mathematical properties for series H|. First, if we use the integral relation:
1
TT = r (b )rr(C) b ) B (b1 + n, C - b1) = _ (b )rr(C) b J b1+n-1 (1 - )c-b1-1d^,
(c)n r (b1) r (c - b1) r (b1) r (c - b1) 7
Re (c) > Re (b1) > 0,
(3.1)
where B(a,b) is Beta function (see e.g. [[8], pp. 9-11], [16] and [[18], p. 26 and p. 86, Problem 1]) defined by
B< b /Jo <"-1(1 - t)b-1dt, (R(a) > 0, R (b) > 0),
(a, b) = | rm, (a,b.C\Z-), (3-2)
the definition (1.4) and the identity
(- 1)m
(f)-m = 7T-rar • (3-3)
(1 f)m
Then we get the following integral representation for
2r (c)
(a, bi, c; x, y) =
r (bi) r (c - bi)
x
i F2b1-1(1 - §2)c-bl-1(l -y;a, 1 + a;1 - b2;-4x (1 -yt2)) d;, (3U)
0
Re (c) > Re (b1) > 0, b2 = 0,±1,±2,... Next, if in (3.4) we employ the result [8]
1
F (",b; c; z) = r (a)T(CC-a) I * °-'(1 - » ^ - )-bd n,
r (a) r (c — a) Re (c) > Re (a) > 0, then formula (3.4) can be rewritten in the more elegant form:
r (c) r (1 — b2)
(a,b1,b2;c;x,y) =
r(b1)r(c-b1) r(f) r (1 -b2- f)
1 1
x||%b1—1n2—1(1 — %)c—b1—1(1 — n)—b2—f(1 — y%)—b2[1 + 4xn (1 — y%)]—^d%dn, 0 0
Re (c) > Re (b1) > 0, Re (1 — b2) > Re (2) > 0.
(3.5)
Again, since
(b1)n = r (c) B (b + nc b )= r (c) 91—n—c
= r (b1) r (c — b1) (b1+n, c — b1) = r (b1) r (c — b1)2
1
x J [(1 + %)b1—1+n(1 — %)c—b1—1 + (1 + %)c—b1—1(1 — %)b1—1+n] d%, (3.6)
0
Re (c) > Re (b1) > 0, we find that H (a, b1, b2; c; x, y)
1c
21-cr (c) 1 + ; ^ -1(1 ; )C-b1-1d; f- (fW^-m-A y (1 + F ) V^ 0(1+;) (1 -;) d;rn^-o—m!—1 ——)
21-cr (c) 1 (1 + ; )C-b1-1(1 ; )b1-1d; f. (f ) 2m (b2) -m m ( 1 y (1 - F ) V^ + ^) (1 - ^) d;m^^o-m!-1--2-) ,
+ -1 c± (c) I n I c-b1-^ ^¿1-1
r (b1) r (c - b1) 0 m=0
Re (c) > Re (b1 ) > 0,
(3.7)
which in view of (3.3) , the result (1.5) and after a little simplification, gives us the integral formula
21+b2-cp (c)
H(a,b1,b2;c;x,y) = r(b1)F(c-b1)
X J (1 + §)b1-1(1 - §)c-b1-1(2 -y -)-b22F^a, 2 + a;1 - b2;2 (xy + xy§ - 2x))
0
21+b2-cp (c)
r(b1)r(c - b1)
J (1 + §)c-b1-1(1 - §)b1-1(2 -y + y§)-b22F^a, 2 + f;1 - b2;2 (xy -xy§ - 2x)) d§,
0
Re (c) > Re (b1) > 0, b2 = 0,±1,±2,...
Similarly, if we use the expression (0)n r(S)
(3.8)
B (ß + n, 8 - ß),
(8)„ r(ß) r(8 - ß)
1
B(x,y) = (1 + A)x|§x-1(1 - §)y-1(1 + A§)-x-yd§, 0
Re (x) > 0, Re (y) > 0, A > -1,
one can show that
(3.9)
H (a,b1,b2;c;x,y) = r(b1)r(c-b1)Q
r(c)(1 + A)b1 1 § b1-1(1 - § )c-b1-1(1 + A§ )b2-c
/'a 1 + a., ^ . -y§ + A(1 -y)§
x [1 -y§ + A (1 -y) §r^Fi (J, 1 - b2; -4x ^+A§ ) d§,
Re (c) > Re (b1) > 0, A > -1, b2 = 0,±1,±2,...
(3.10)
Moreover, by using the following well-known integral representation for Beta function (see, for example [[18], p. 86, Problem 1]):
b
B (^y)=(b(b--c^-c1)7 (b - §)y-1(§ - a)x-1(§ - c)-x-yd§, (311)
Re (x) > 0, Re (y) > 0, c < a < b, it is not difficult to show that
r(c) (ß - Y)b1 (a- Y)c-b1 ß,fi §\c-b1-1 /§ xb1-1
H (a, b1, b2; c; x, y) =-^—" . 1 / (ß - § )c-b1-1(§ - a)
2V ; r(b1)r(c-b1)(ß - a)c-b2-1 ' V ^ V ;
a
X (§ - Y)b2-c[(ß - a) (§ - Y) - y (ß - Y) (§ - a)]-b2 (3.12)
xF (a1 + a b . (ß - a)(§ - y) - y (ß - Y)(§ - a) \ d§
X y2 , 1 - b2; -4x-(ß - a )(§ - y)-) d §,
Re (c) > Re(b1) > 0, Y< a < ß, b2 = 0;±1,±2,...
Further, in view of [[16], p. 26, Equation 49]):
n 2
B (x,y) = 2 J (sin2%)x-2 (cos2%)y-1 d%, Re x > 0, Re y > 0, (3.13)
we obtain
H (a,b„*c;x,y) = bJ ^n2«1 (cos2«2
r(bi) r(c - bi)
0
x(1 -ysin2%)-b2d% £ (a)2m(b2)-m [x(1 -ysin2%)]m,
m=0 m'
Re (b1) > 0, Re (c - b1) > 0, which after a little simplification, gives us the result
n 2
H2 (a,b1,b2;c;x,y) = 2Î(t b[)/ (s'n2«)b'-2 (cos2%)c-'1-2
0
x(1 -ysin2%)-b22F^2,^; 1 -b2. — 4x(1 -ysin2%)) d%, Re (b1) > 0, Re (c-b1) > 0, b2 = 0,±1,±2,... = 0,±1,±2,...
Similarly, if we consider the relations
r(b1)r(c - b1)(b1),
r(c) (c)n
= B (b, + n, c - b,)
and
(3.14)
= 2(1 + A)bi+« / (sin2«)bi+n-1 (cos2«)c-b1-2d«
2(1 +A) 0 (1 + W«)n+c ^ (3.15)
Re (b, + n) > 0, Re (c-b,) > 0, A >-1,
= 2Ab1+n } (sin2«)b1+n-2 (cos2«)c-b1-2
0 (cos2« + A sin2« )n+c ^, (3.16)
Re (b1 + n) > 0, Re (c - b1) > 0, A > 0,
= 2a 2b1+2nß 2c-2b^ sin2b1+2n-1 ç cos2c-2b1-1ç [a 2sin2ç + ß 2cos2ç p-crfç, (3 1?) 0
Re (c) > Re (b1) > 0, a > 0, ß > 0,
we can easily derive the following integral representations
H (a,b1,b2;c;x,y) = ™ + A£ / ^^2 ^f"1 2 ( , ^ 2; ; ,y) r(b1)r(c — b1) 0 (1 + asin2%)c—
x [1 + Asin2% — y (1 + A) sin2%] —b2 (3.18)
(a 1 + a , 1 + Asin2% — y(1 + A)sin2%^
X2F^2 — b2; —4x--Jd %,
Re (b1) > 0, Re (c — b1) > 0, A > —1,
2r (c) A b
(a,bi,c;x,y) =
r (bi) r (c - bi)
2 ^)b1-^ (C°S2*)C-b1-^ (cos2^ + Asin2^ -yAsin2^)-b2 „ ^
£ + A sin2^)c-b^ j (3.19)
0 (cos2
^ fa 1 + a 1 , , cos2^ + A sin2^ - yA sin2^', _
X2F1 2-b2;-4x-2. . 2e-K,
\2 2 cos2e + A sin2e
Re (b1) > 0, Re (c - b1) > 0, A > 0,
and
H (a, b1, b2; c;x,y) = ^^P)^^ / sin^Vos^^p
X [a2sin2p + p2cos2p]b2—c [(1 — y) a2sin2p + p2cos2p]—b2 (3.20)
^ fa 1 a 1 , , (1 — y) a 2sin2p + P 2cos2p\ ,
X 2F1 2, 2 + 2; 1 — b2; —'^-2 ' 2 p2 2— dP,
\2 2 2 a2sin2p + p2cos2p /
Re (c) > Re (b1) > 0, a > 0, p > 0, b2 = 0,±1,±2,...
respectively.
Finally, in view of the well-known integral representation for Gamma function [8]:
pc
r(z) = e—iz—1di, Re(z) > 0, (3.21)
0
we have
oo
(a)2m = = rW)l e-lea+2m-1d e, Re (a) > 0. (3.22)
Thus for the Horn-type function H|, we can show that
cc
H (a,b1,b2;c;x,y) = ^r) je—%%a—1H4 (b2,b1;c;y,x%2) d%,Re(a) > 0, (3.23)
where
H (/,ß;«;y)= £
m,n=0 (u)mw-"'
is Horn confluent series [[8], section 5.7.1]. Similarly, by considering the Horn confluent series
H5 (a ; p ; x, y)= £ (fex-y",
m,n=0 (ii)mm-n-
we can easily derive the following integral representation
CC C
H| (a,b1,b2;c;x,y) =j je-%e-n%a-1nb1-1(b2;c;yn,x%2) d%dn,
1 0 0 (3.24)
Re (a) > 0, Re (b1) > 0.
4. Expansion, analytic continuation and transformation formulas
First, in [4] the authors introduced the following inverse pair of symbolic operators: Hx (a, c) = £ Of-xli, H (a, c) = £ ^ f-^, (4.1)
i=0 (c);;! i=0 (1 - a- W
where 5x = xJx. Then
H| (a,b1,b2;c;x,y) = £ i^iM(-1)iyi £ ^^ ;)n-^yn,
;=0 (c)ii! m,p=0 m^!
by using (3.3) and (1.5), gives us the expansion formula: H2 (a, b1, b2; c; x, y)
M N-b2 f (c-b1)f(b2)i ( y Y' (a a + 1 1 , . , , 1 A (4.2)
=(1 -y) b2 ;£0( (c));-i u-ij 2f y2, - b2- i; -4x(1 -yV .
On other hand, we have
(1 - y)-c2F^a, 2 + 2;1 - b2; -4x (1 - y)) = H (c,bx)H| (a,b1,b2;c;x,y). Therefore, from the second operator in (4.1), we get
(1 - 2, a + - b2; —4x (1 - y)
- £—(1—Ь1).(С).Л—y H (a'b1'b2+i; c+i; x y)
(4.3)
Secondly, starting from the series
(2c — 2b2)2m(b2)
(2c — 2b2 ,bbb2;c;x,y) _ £ ^-^F (bb b2 — m;c;y), (4.4)
m-0 m'
taking to account that the Gaussian hypergeometric function 2F1 has the relation [[8], p. 108-109]
F (a, b; c;x) _ Г^ГУ — 1 (1 — '[a, c — b; a — b + 1; (1 — x)—1
i (b) i (c a) l ^ (4 5)
r(c))r(a — Ь) (1 — x)—b2F1 [b,c — a;b — a + 1; (1 — x)—1l , | arg(1 — x)| < n,
(a) (c - b)
and by employing (3.3) and (1.5), we obtain by means the routine calculations
(2c — 2b2, b1, b2; c; x, y)
= r (c) r (b2 — b1) ( b1 c (c —b2)m+n( 1 + c —b2)m(b1 )n (_4x)m (
r(b2)r(c —b1)( y) m:£=0 (1 — b2 + b1)m+nm!n! ( ) W— y
+ r (c) r (b1 — b2) (1 y) — b2 c (2 + c —b2)m(c —b1)n(b2)n—m ( 4x)m(1 y)m ( 1
+ r (b1) r(c —b2)(1— y) m^ (1— b1 +b2)n—mm!n! (—4x) (1— y)
(4.6)
Now, if we consider the definitions of the Appell's function F1 and the Horn function G2 then from relation (4.6) (See [8]), we infer the following
(2c — 2b2, b1, b2; c; x, y)
r (c)r (b2 b1^ , ^ L u.1 , „ u U.I u tu. ^ 1
(1 -y) b1 FW c b2; — + c-b2,b1 ; 1 -b2 + b1;-4x,
"r (b2) r (c - b^v " 1 V 2 '1 - yj (4.7)
^(c) rib1 -b2) (1 -y)-b2gJ1 + c- b2,c-b1,b2,b1 - b2;4x(1 -y), - 1
r (bi) r (c - ^^ 2' ° 2' 1 2' v 1 - y
Alternatively, starting from (4.4), employing the relation
2F1 (a, b; c;x) = (c^-^ - b) x-a2F [a, a - c + 1; a + b - c + 1;1 - x-1]
+ r (cr^r c) xf-C(1 - x)C-f-b2F1 [c - f, 1 - a; c - f - b + 1;1 - x-1], (4^8)
| argx| < n,
and following the method of the derivation of equation (4.6), we can establish the analytic continuation formula:
nu u u \ r (c) r(c — b1 — b2) —b1
H2 (2c — 2b2, b1, b2; c; x, y) = r (c — b1)r (c — b2)y 1
xH^ c — b1 — b2;1 + c — b2, b1,1 + b1 — c;1 — b2; —4x, -y+
+ r (c) r (b1 + b2 — c) vb1—c(i y)c—b1
+ r (b1) r (b2) y ( y)
xF3 ^c — b2, c — b1, c — b2 + 1, 1 — b1; 1 — b1 — b2 + c; —4x (1 — y), 1 — where F3 is Appell's double series defined by
(a1)m(b1)n(a2)m(b2)n
(4.9)
F3 (f1,b1,f2,b2;c;x,y) = £ ^ci^mn^^y'
m,n=0 (c)m+nm!^!
Next, if in (3.4) let a = 2 — 2b2 and by using the identity
1 + 4x (1 — y% ) = 1 + 4x — 4xy% = (1 + 4x)( 1 —
40
we get
r(c)
H(2 - 2b2,bi,b2;c;x,y) =
r(bi) r(c - bi)
- 1 ( 4 § \- — (4.10)
X(1 + 4x)-3-^| <f1-1(1 - § )c-bl-1(1 - )-b2(l - 2 .
0
Now, if we use (3.2) then we will obtain a transformation relation in the form
2b2-3 / 3 — 2bo 4xv \
H(2-2b2,¿1,b2;c;x,y) = (1 + 4x)~fA by,b^—c;y, y+^xJ. (4.11)
Further, if we start from (4.2), use (1.1) and simplify, we obtain
U U ^ n \-¿2 f (c - b1)i(b2)i(a)2j ( y Yr n
2 (a,b1,b2; C;X,y) = (1 - y) 2 ^0 (c)i(1 - ¿2 - i) jiljAy^Tj [-X (1 - y)]j
(—1)j(b)
which on making use the identity (1 - b - z)j = ((b).v/i, gives us
H (a, b1, b2; c; x, y) = (1 - y)-b1 c - b1, b2; c; x (1 - y), y-^ . (4.12)
5. Generating relations via operational identities
The principle of the operational techniques provides a powerful and flexible means to deal with hypergeometric functions of one, two and multiple variables. In fact, an appropriate combination of methods, relevant to operational calculus and to special functions, can be a very useful tool to establish and treat operational identities for hypergeometric functions. In this regard, the following two formulas are the well-known consequences of the derivative operator DDx and the integral operator DD-1 [15]:
fcx" = fg-Sr x" (5.1)
DD-n xa = r(q+1) xa+n (5 2)
Dx x = r(a+n+1) x , (5.2)
where a e C\{-1,-2,...},n e NU{0}. Based on the operational relations (5.1) and (5.2), we first prove the following Lemma.
Lemma 1. Let Re(a) > 0, Re(b1) > 0, Re(b2) > 0, Re(c) > 0 and , Re(e) > 0, then H (a, b1, b2; c;x,y) |xa-1yb2-1ic-1)
1 b r . (5.3)
= (1 -yt-1D)-1DDyy)-b1 x exp [xZD2x2DD-1y-1^xa-1yb2-1ic-^ , H (a,b1,b2;c;x,y) |xe-1yb2-1tb2-1)
r , (5.4)
= 2F1 (a, 2 + 2;e;4xZDxxZD-1t-1) x 2F1 (bbb2;c;DD-14i) {xe-1yb2-1tb2-1} .
Proof. Denote, for convenience, the right-hand side of assertion (5.3) by I. Then as a consequence of the binomial theorem and the exponential function ex, it is easily seen that:
I = II ^ yst-V (4-Y-1) (42r*fl+2r-1) fer^--1). (5.5)
s=0r=0 V 7
Upon using (5.1), (5.2) and considering the definition (1.4), we are led finally to the left-hand side of the assertion (5.3). The proof of the operational representation (5.4) runs parallel to that of (5.3) then we skip the details.
Now, we will explore the formal properties of the operational identities (5.3) and (5.4) to derive some generating functions for H|. First of all, in the identity (5.3) put b1 = -m, m e N U{0}, multiply throughout by tm/m! and then sum to get the generating relation
ca-y2-iiC-i 1 I H (a, -m,c;x,y)
um
m=0 m! (5.6)
= exp [(1 — yt—1Z)—1Z)yy) w] x exp [xZ^x2^-V1] {xfl—1yb2—1tc—^ .
In the same manner, from the operational identity in (5.3) one can derive the following generating function
a—1yb2—1tc— 1 I £ (a, —m,b2;c;x,y) wm
J m=0 (5.7)
= (1 — u + yt-1DDyyw)—1 x exp [x^^y-1 ] {x^T^-V-1} .
Further, to obtain more generating functions our starting point will be some of the bilinear generating functions presented in [[18], Chapter 5]. First, let us consider the bilinear generating function [[18], p. 308 (ii)]
n 7 n
I r (-1)r2Fi (-, a;ß;x) 2F1 (-n + r, r; 5;y) r=0 V J (5.8)
x"3F2 (-n, 1 - ß - n, y; 1 - a - n, 5;
(P )n
If in (5.4), we put a = -2r and b1 = -n + r, {r,n} e N U{0}, multiply throughout by ^ n ) (-1)r, take the sum of both sides and then compare the resulting equation with the generating function (5.8), we obtain
n ' n
I r (- 1)rH2 (-2r, r - n, b2; c; x, y)
r=°/1 \ (5.9)
= (T-iS(-4x)n3F2 (-n, ^,^ + 2;e - n + 1,c;-.
Similarly, by considering the following bilinear generating functions [[18], p. 299(5.3) (11)]
I 2F1 (p - n, a; p;x) 2F (A + n, 7; 5;y) vn
r=0 n (5.10)
= (1 - v)-AFm(y,a,a,A,p,A;5,p,p,x,vxv1).
(5.11)
and [[18], p. 294(5.3) (1)]
£ (n-r2F1 (A + n,a;p;x)2F1 (-n,7;5;y) vn = (1 - v)-AF2 (A,a,7;P,5,-^
r=0
we can establish the results
£ ^(p - n, A + n, b2; c;x, y) vn
= (1 - v)-AF^ (1 (p - n + 1), A, 2p,b2;c;4x, ^, ). (5.12)
where (see[[17], p. 77, Series (11d)])
„(3), , f (a1 )m+n (a2)n+p (a3)m (b)p-m-„ xmynzp 1
F1W(a1,a2,b; c;; x,y, z)=m,£=0-(^-mrnrpy, (5.13)
and
£ ^H (2A + 2n, -n,b2;c;x,y) vn
r=0 n! (5.14)
= (1 - v)-AH1 (b2, A, A + n + 2;c;-vy, ^),
where
b X V (a)m-n (b)m + n (c)n xmyn
H1(a,b,c;e;x,y)= £ ---—, (5.15)
is the Horn series defined in [[17], p. 24, (9)], respectively. Finally, it is important to note that the operational representations (5.3) and (5.4) can, in turn, be used to state other needed properties of hypergeometric series H|. For instance, if we let
M = (1 - yt-1DD --1DD yy)
then from relation (5.3), we can state that
[1 - Mx exp [xDD2x2DD-1y-1] |xfl-1yb2-1tc-1}
"" ' ^ (1 -yt-14-1DDyy)S X exp [xDd2x2DD-1y-1] |xa-1yb2-1tc-1|
s=0
which in view of (5.3), gives us the result
[1 -M]n x exp [x^x2^-1] {x^V2-1^-1}
= Ê^) (-1)SH| (a, -s, b2; c; x, y.) (5.16)
On other hand, we have
[1 -M]n x exp [xD)2x2D)-1y-1] |xa-1yb2-1ic-1}
= [1 - (1 - yt-14-1DV)r x exp [x42x2DVy-1] {xa-V2-1tc-1}
43
= (yt-14-1Dyy)n x exp [xD^DVV1] {xf-1yb2-y-1} = £ ^Zi— (DD-ntc-1) (D2mx§+2m-1)
m=0
which in view of the formulas (5.1) and (5.2)and considering the definition of Gaussian hypergeomtric function 2F1; we get
[1 -M]n x exp [xD2x2D-1y-1] {xfl-1yb2-1tc-1} = yn(C2)n2F1 (f,§ + 1;1 -b2-n;-4x), n = 0,1,2,... (5.17)
(c)n
Hence from (5.16) and (5.17), we obtain the following interesting summation formula:
n
n 1 (-1)sH2* (a, -s, c;x,y) j=o\ s /
yn(b2)n
(C)n
Acknowledgments
2F1 (a,a + 1;1 -b2-n;-4x) , {n = 0, l,2,...}. (5.18)
We would like to thank the anonymous reviewer for valuable suggestions, which made the present paper more readable.
References
[1
[2
[3 [4 [5 [6 [7 [8 [9 [10 [11
Aomoto K., "On the structure of integrals of power products of linear functions", Sci. Papers, Coll. Gen. Education, Univ. Tokyo, 27 (1977), 49-61.
Appell P., Kampe de Feriet J., Fonctions Hypergeometriques et Hyperspheriques: Polynomes d'Hermite, Gauthier-Villars, Paris, France, 1926.
Borngasser L., Uber hypergeometrische funkionen zweier Veranderlichen, Dissertation, Darmstadt, 1933.
Choi J., Hasanov A., "Applications of the operator H(a,b) to the Humbert double Hyper-geometric functions", Comput. Math. Appl., 61 (2011), 663-671.
Carlson B. C., "Appell's function F4 as a double average", SIAM J. Math. Anal., 6 (1975), 960-965.
Carlson B. C., "The need of a new classification of double hypergeometric series", Proc. Amer. Math. Soc., 56 (1976), 221-224.
Erdelyi A., "Transformations of hypergeometric functions of two variables", Proc. Roy. Soc. Edinburg Sect. A, 62 (1948), 378-385.
Erdelyi A., Magnus W., Oberhettinger F., Tricomi F. G., Higher transcendental functions. V.I, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1953.
Exton H., Multiple hypergeometric functions and applications, Ellis Horwood Ltd., Chich-ester, New York, 1976.
Gelfand I. M., Gelfand S. I., "Generalized hypergeometric functions", Dokl. Akad. Nauk. SSSR, 228 (2) (1986), 279-283.
Heckman G. J., Opdam E. M., "Root systems and hypergeometric functions I", Comp. Math, 64 (1987), 329-352.
[12] Horn J., "Hypergeometrische Funktionen zweier Vernderlichen", Math. Ann., 105(1) (1931), 381-407.
[13] Horn J., "Uber die convergenz der hypergeometrischen reihen zweier und dreier Veränderlichen", Math.Ann., 34 (1889), 544-600.
[14] Horn J., "Hypergeometrische Funktionen zweier veränderlichen", Math. Ann., III (1935), 638-677.
[15] Miller K.S., Ross B., An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley and Sons, Inc., New York, 1993.
[16] Srivastava H.M., Choi J., Series Associated with the Zeta and Related Functions, Kluwer Academic Publishers, Dordrecht, Boston and London, 2001.
[17] Srivastava, H. M., Karlsson, Per W., Multiple Gaussian hypergeometric series, Ellis Hor-wood Series: Mathematics and its Applications, Ellis Horwood Ltd., Chichester, 1985.
[18] Srivastava, H. M., Manocha, H. L., A treatise on generating functions, Halsted Press, New York, Brisbane and Toronto, 1984.
References (GOST)
[1] Aomoto K. On the structure of integrals of power products of linear functions, Sci. Papers, Coll. Gen. Education, Univ. Tokyo 27, 1977, pp. 49-61.
[2] P. Appell and J. Kampe de Feriet. Fonctions Hypergeometriques et Hyperspheriques: Polynomes d'Hermite, Gauthier-Villars, Paris, France. 1926
[3] Borngasser, L. Uber hypergeometrische funkionen zweier Veranderlichen, Dissertation, Darmstadt. 1933.
[4] Choi J. and Hasanov A. Applications of the operator H(a,b) to the Humbert double Hypergeometric functions, Comput. Math. Appl. 61, 2011, pp. 663-671.
[5] Carlson B. C. Appell's function F4 as a double average, SIAM J. Math. Anal. 6, 1975, pp. 960-965.
[6] Carlson B. C. The need of a new classification of double hypergeometric series, Proc. Amer. Math. Soc. 56, 1976, pp. 221-224.
[7] Erdelyi A. Transformations of hypergeometric functions of two variables Proc. Roy. Soc. Edinburg Sect. A 62, 1948, pp. 378-385.
[8] Erdelyi A.; Magnus W.; Oberhettinger F.; Tricomi F. G. Higher transcendental functions Vol. I., McGraw-Hill Book Company, Inc., New York-Toronto-London. 1953.
[9] Exton H. Multiple hypergeometric functions and applications, Ellis Horwood Ltd., Chichester, New York. 1976.
[10] Gelfand I. M. and Gelfand S. I. (1986) Generalized hypergeometric functions, Dokl. Akad. Nauk. SSSR 228 (2), 1986, pp. 279-283.
[11] Heckman G. J. and Opdam E. M. Root systems and hypergeometric functions I, Comp. Math. 64, 1987, pp. 329-352.
[12] Horn J. Hypergeometrische Funktionen zweier Ver?nderlichen, Math. Ann., 105(1), 1931, pp. 381-407.
[13] Horn J. Uber die convergenz der hypergeometrischen reihen zweier und dreier Veranderlichen, Math.Ann., 34, 1889, pp. 544-600.
[14] Horn J. Hypergeometrische Funktionen zweier veranderlichen, Math.Ann. III, 1935, pp. 638-677.
[15] Miller K.S. and Ross B. An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley and Sons, Inc., New York. 1993.
[16] Srivastava H.M. and Choi J. Series Associated with the Zeta and Related Functions. Kluwer Academic Publishers, Dordrecht, Boston and London. 2001.
[17] Srivastava, H. M.; Karlsson, Per W. Multiple Gaussian hypergeometric series, Ellis Horwood Series: Mathematics and its Applications, Chichester: Ellis Horwood Ltd. 1985.
[18] Srivastava, H. M. and Manocha, H. L. A treatise on generating functions, Halsted Press, New York, Brisbane and Toronto. 1984.
Для цитирования: Hasanov A., Saad M. G. B., Ryskan A. Some properties of Horn type second order double hypergeometric series // Вестник КРАУНЦ. Физ.-мат. науки. 2018. № 1(21). C. 32-47. DOI: 10.18454/2079-6641-2018-21-1-32-47
For citation: Hasanov A., Saad M. G. B., Ryskan A. Some properties of Horn type second order double hypergeometric series, Vestnik KRAUNC. Fiz.-mat. nauki. 2018, 21: 1, 32-47. DOI: 10.18454/2079-6641-2018-21-1-32-47
Поступила в редакцию / Original article submitted: 09.02.2018 В окончательном варианте / Revision submitted: 14.03.2018
Vestnik KRAUNC. Fiz.-Mat. Nauki. 2018. no.1(21). pp. 32-47. ISSN 2079-6641
DOI: 10.18454/2079-6641-2018-21-1-32-47
УДК 517.58
НЕКОТОРЫЕ СВОЙСТВА ГИПЕРГЕОМЕТРИЧЕСКОГО РЯДА ГОРНА ВТОРОГО ПОРЯДКА
А. Хасанов1, М. Г. Б. Саад2, А. Рыскан3
1 Институт математики, г. Ташкент, ул. Мирзо Улугбека, 8, 700170, Узбекистан
2 Отдел математики, Аденский университет, г. Аден, Кохрмакссар, 6014, Йемен
3 Институт математики, физики и компьютерных наук, КазНПУ имени Абая, Алматы,
ул. Толе би, 86, 0500012, Казахстан
E-mail: anvarhasanov@yahoo.com
В работах Горн [1931, Hypergeometrische Funktionen zweier Veranderlichen, Math. Ann., 105 (1), 381-407], (исправления в книге Борнгассера [1933, Uber hypergeometrische funkionen zweier Veranderlichen, Диссертация, Дармштадт]) были определены и исследованы десять гипергеометрических рядов двух переменных второго порядка. Исследуя ряды Горна, мы заметили существование гипергеометрических двойных рядов Я|, аналогичных двойному ряду Горна Я2. Основная цель настоящей статьи это представить дальнейшие шаги исследования математических свойств и представлений, относительно аналогичных гипергеометрических двойных рядов H2. Действительно, воодушевленные важной ролью функций Горна в нескольких разнообразных областях физики и вкладом в унификацию и обобщение гипергеометрических функций, мы составляем систему уравнений в частных производных, интегральные представления, формул разложения, аналитическое продолжение, формулы преобразования. А также обсуждаются связи результатов, представленные в этой статье с уже известными.
Ключевые слова: гипергеометрическая функция Гаусса, двойные ряды Хорна, уравнения в частных производных, интегральные представления, преобразование, производящие функции
© Хасанов А., Саад М. Г. Б., Рыскан А., 2018
