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EDN: BFKSUT УДК 517.5
A Note on Two General Reduction Formulas for the Srivastava-Daoust Double Hypergeometric Functions
Musharraf Ali*
Department of Mathematics, G. F. College Shahjahanpur-242001, Uttar Pradesh, India
Harsh Vardhan Harsh
Department of Mathematics The ICFAI University Jaipur-302031, Rajasthan, India
Arjun K. Rathie*
Department of Mathematics Vedant College of Engineering and Technology Rajasthan Technical University Bundi-323021, Rajasthan, India
Received 23.04.2023, received in revised form 18.08.2023, accepted 14.09.2023
Abstract. The aim of this note is to provide two new and general reduction formulas for the Srivastava-Doust double hypergeometric functions. A few interesting special cases have also been given.
Keywords: hypergeometric functions, Humbert double hypergeometric functions, Appell functions, Srivastava & Daoust double hypergeometric function, beta integral method.
Citation: M.Ali, H.V. Harsh, A.K. Rathie, A Note on Two General Reduction Formulas for the Srivastava-Daoust Double Hypergeometric Functions, J. Sib. Fed. Univ. Math. Phys., 2024, 17(1), 48-54. EDN: BFKSUT.
1. Introduction and results required
We start with the definition of generalized hypergeometric function pFq with p numerator and q denominator parameters as [11]
F
pFq
ai, . .., Op Ь1, ..., bq
^(°i)n••• K)n ZI (1 n
¿0 b)n • • b)n n! ' (-)
where (a)n denotes the well-known Pochhammer symbol (or the raised or the shifted factorial, since (1)n = n!) defined for any complex number a by
ia(a +1) ... (o + n - 1), n G N Г(о + n) ^^
(a)n =<. n = s , (1.2)
11, n = 0 1(a)
where r is the well-known Gamma function. For convergence conditions (including absolute convergence) and properties of this function, we refer standard texts [2,11].
* [email protected] https://orcid.org/0000-0001-9791-3217 [email protected] ^ [email protected] © Siberian Federal University. All rights reserved
It is not out of place to mention here that the vast popularity and immense usefulness of the hypergeometric function 2F1 (which is a special case of pFq for p = 2 and q =1) and the generalized hypergeometric function pFq in one variable have inspired and stimulated a large number of researchers to study hypergeometric functions of two and more variables. In this contexts, serious and very significant study of the functions of two variables initiated by Appell [1] who introduced the so-called functions Fi, F2, F3 and F4 named in the literature, the Appell functions which are the natural generalizations of 2F1 and pFq. Also their confluent forms were studied by Humbert [16,17]. A complete list of these functions can be seen in the standard text [7] and also in [3,4].
Later on, the Appell functions F1, F2, F3 and F4 and their confluent forms were further generalized by Kampe de Feriet [1], who introduced a more general function in two variables. The notation for this function was subsequently abbreviated by Burchnall and Chaundy [5,6]. However, in our present investigations, we recall here the definition of a more general function in two variables (than the one defined by Kampe de Feriet) in a slightly modified notation which is due to Srivastava and Panda [18, Equ.(26), p.123] defined as follows
F
H:A;B G:C;D
(hH) :
(sg):
(«a); (cc );
(bB );
(dn );x,y
CO CO
((hH))m+n((aA))m((&B))n x™ y^ ^3)
m=0 n=0
((gG ))m+n((cc ))m((dn ))n m! n!
where (hH) denotes the sequence of parameters (hi, h2, ■ ■ ■ Pochhammer symbol
((hH))n := (hi)n ■ ■ ■ (hH)n,
, hH ) and for n G N0, define the
where, when n = 0, the product is understood to reduce to unity. The symbol (h) is a convenient contraction for the sequence of parameters h1, h2, ■ ■ ■ ,hH and the Pochhammer symbol (h)n is the same as defined in (1.2). For details about the convergence for this function, we refer to [16].
The Srivastava-Daoust generalized Kampe de Feriet hypergeometric function of two variables initially introduced in [13,14] will be defined and represented in the following manner:
SA:B;B' SC:n;D'
SA:B;B' — SC:D;n'
[(a): Mj: [(b): # [(&'): n xy
[(c): S,ej: [(d): Vj; [(d') : n'j; X y
EE
rj=i r(aj + m9j + n4>i ) rj=i r(bj + mij )Y\B=i r(bj + n^ ) ^ y,n m=on=o nC=i r(Cj + m5j + nej) nD=i r(dj + mnj) nf= i r(dj + mVj) m! n!
where, for convergence
(1.4)
C n A B
1 + £ Sj ^ Vj -¿2 0j - £ $j > o,
j=i j=i j=i j=i
C n' A B'
1 + £ j +Y, j - £ h - £ > 0.
j=i j=i j=i j=i
A detailed account of the above function can be found in the research paper [15] and in the text [17].
We also give below the definitions of the Humbert function (confluent forms of the Appell functions) in the following manner ( [3-6,8,10,17]).
oo oo
(b)k wk z
k ..m
$3(b; c; w, z) = J2 E ^ z-i I
(c)k+m k! m!
k=0 m=0 k+m
(1.5)
which converge absolutely at any w, z G C.
By employing the following generalizations of the Kummer's summation theorem due to Rakha and Rathie [12] viz.
2F1
b
a - b + n + 1 '
2n-2br(b - n)r(a - b + n +1) r(b)r (a - 2b + n +1) '
a+k+n+1
¿<-1>*( n)
k=0 v 7
r(
2
-b
r(a+k —n+1^
and
2 F1
a - b - n + 1
1
-2b—nT
T(a - b - n +1) ^ r (a - 2b - n +1) X ^
k=0
n
^ r( k
a+k—n+1 2
-b
r(
a+k—n+1^ 2 ;
(1.6)
(1.7)
Very recently, Brychkov et al. [4] established the following general reduction formulas for the Humbert functions $3 viz.
(a) For n = 0,1, 2,..., the following result holds true:
$3( b; ^ + 1; z, -/
(-2)n
^(-1)H k}X
k=0
n
b+n+2k+2 ^ r ( b—3n+2k+2 y 3F4
+
4 8z
1 b+n+2k+2 2 — 2k — b+3n 1, 4 , 4
n+1 n +1 b+n+2 b+n I 1 ; 2 , 2 +1, 4 , 4 +1
; - z
+
^ 5+„+2fc + 1)
(n + 1)(b + n + 2) r (b—3n+2ky
3F4
1 b+n+2k + 1 1 _ b—3n+2k 1, 4 + 1, 1 4
n +1 n+3 b+n + 1 b+n+6
2 + 1, 2 , 4 + 1, 4
; - z
(1.8)
x
2
b
(b) For n = 0,1, 2,..., the following result holds true:
$3 b
b;
b
+ 1; z, -z2 =2
)=*-£( k) {=F3
b—n+2k+2 2 —2k —b+n
4 , 4
1 b—n+2 b—n + 1 2, 4 , 4 + 1
;z
+
2(b - n + 2k) z
+--1-To- 2F3
b - n + 2
b—n+2k
b—n+2k
4 +1, 1 - 4
3 b—n + 1 b—n+6 2, 4 + 1, 4
(1.9)
In addition to this, the Beta function B(a, b) is defined by the first integral and known to be evaluated as the second one as follows:
B(a, b) =
/ za—1(1 - z)b—1dz, (Re(a) > 0, Re(b) > 0)
r(a)r(b)
r(a + b) ,
(a, b e C \ Z—)
(1.10)
2
z
Recently, Krattenthaler and Rao [9] made a systematic use of the so-called Beta integral method, a method of deriving new hypergeometric identities from old ones by mainly using the beta integral in (1.10) based on the Mathematica Package HYP, to illustrate several interesting identities for the hypergeometric function and Kampe de Feriet functions.
Thus, the aim of this note is to provide certain new and general transformation formulas for the Srivastava-Daoust double hypergeometric functions with the help of the general reduction formulas (1.8) to (1.9). The results are established by employing the beta integral method. A few results obtained earlier by Wei et al. [19] follow special cases of our main findings.
The results presented in this note are simple, interesting, easily established and may be useful (potentially).
2. Transformation formulas
The new and general transformation formulas for the Kampe de Feriet functions to be established in this note are given in the following theorems.
Theorem 2.1.
S
1:1;0 2:0:0
[e : 1 ,2] :
[d : 1 ,2] , [ ^ + 1:1 ,1]:
[b : 1]; -
x< 5F6
+
1 b+n+2k+2 2-2k-b+3n e e+1 1 4 , 4 ' 2 ' 2
n+1 n I 1 b+n+2 b+n I 1 d d+1 2 , 2 + 1 4 , 4 + 1 2, 2
8ue r ( b+n4+2k +1)
d(n + 1)(b + n + 2) r (b—^+2k)
(—2)n
k=0
(—1)
■Ö
X 5F6
+
1 b+n+2k +1 1 _ b—3n+2k e+1 e + 1
1, 4 + 1, 1 4 , 2 , 2 +1
n + 1 n+3 b+n I 1 b+n+6 d+1 d + 1
2 + 1 2 , 4 + 1 4 , 2 , 2 +1
(2.1)
Theorem 2.2.
1;1;0 S2:0;0
[e : 1, 2]
[b : 1]; —
[d : 1, 2], [ — + 1:1,1]: —;
x{ 4F5
b—n+2k+2 2 —2k —b+n e e+1
4 , 4 , 2, 2
1 b—n+2 b—n + 1 d d+1
2 , 4 , 4 + 1, 2 , 2
;u
+
k=0
0 =
2ue(b — n + 2k) F + d(b — n + 2) 4 5
b—n+2 k +1 1 _ b—n+2 k e+1 e +1
4 +1, 1 4 , 2 , 2 +1
3 b—n + 1 b—n+6 d+1 d +1 2 ' 4 + 1 4 ' 2 ' 2 +1
■
(2.2)
Proof. For the derivation of the result (2.1) asserted in the theorem 2.1, we proceed as follows. First of all replacing z by zu in the reduction formula (1.8), we have
$3 b;
b+n
+ 1; uz, — (uz)2 I =
)-—F »->*( >
r(
b+n+2k+2 N 4 ,
+
^ b—3n+2k+2 ^
8uz
3F4
p ( b+n+2 k +
(n +1)(b + n + 2) r (b—3n+2kY
1 b+n+2k+2 2 —2k —b+3n '
n+1 n4 + 1 b+n+24 b+n + 1 ; —(uz) 2 , 2 + 1, 4 , 4 +1 .
+
(2.3)
x 3F4
b+n+2k b—3n+2k
1-4--r 1, 1--4--• _(uz)2
n + 1 n+3 b+n + 1 b+n+6 ; (uz)
2 + 1' 2 ' 4 + 1' 4
Now, multiply the left-hand side of (2.3) by ze 1(1 — z)d e 1 and integrating with respect to z over the interval [0,1], we have
L.H.S
f ze—1(1 — z) 0
d—e—1$3(b;
b+n 2
+ 1; zu, —(zu)2)dz■
Expressing the Humbert function $3 as a double series with the aid of its definition (1.5), change the order of integration and summation (which is easily seen to be justified), we have
00 00
L.H.S = ((+ik(, 1)muk+2m J ze+k+2m—1(1 — z)d—e—1
k=0 m=0 V 2
№ + ^ k+m k! m\j 0
dz■
u
x
2
u
2
u
2
u
u
n
2
2
u
2
u
2
x
Evaluating the beta integral and using the result
r(a + k + 2m) =
ra =
(a)
k+2m
and after some simplification, summing up the series by interpreting with the help of (1.4), we have
L.H.S
r(e)r(d - e)
S2:0;0
[e : 1, 2]
[b : 1]; -
[d : 1, 2], [^ + 1:1,1]: -;
(2.4)
Similarly, multiply the right-hand side of (2.3) by ze 1(1 - z)d e 1, and integrating with respect to z over the interval [0,1] and proceeding as above, we have
R.H.S
_r(e)r(d - e)(-2)n
r(d)
n!
¿^K n)
k=0 V 7
X 5^6
1 b+n+2k+2 2 —2k —b+3n 1, 4 , 4 ,
n+1 n + 1 b+n+2 b+n +1 d d+1 2 , 2 + 1' 4 , 4 + 1' 2, 2
2 ■
e+1 2
d d+1 2' 2
+
+
8ue r(b++k
1
(2.5)
d(n + 1)(b + n + 2)r (b-3n+2k)
x 5F6
1 b+n+2k +1 1 _ b—3n+2k e+1 e + 1
1, 4 +1, 1 4 , 2 , 2 + 1 . .
n + 1 n+3 b+n + 1 b+n+6 d+1 d +1 ;
2 +1' 2 , 4 +1' 4 , 2 , 2 +1
Finally, equating the equations (2.4) and (2.5), we arrive at the result (2.1) asserted in the Theorem 2.1. this completes the proof of the result (2.1).
In exactly the same manner, the result (2.2) asseerted in the Theorem 2.2 can be established.
□
3. Corollaries
In this section, we shall mention some of very interesting results of our main findings. Corollary 3.1. In Theorem 2.1 or 2.2, if we take n = 0, we get the following result:
S.
1:1;0 2:0;0
[e : 1, 2] [d : 1, 2], [2 + 1:1,1] :
: [b : 1]. -
3F4
2 — b 4 , 2 ' b 2 , 4
e+1 . 2
422
1b + 1 d d+1 ; u
2,4 + 1 , 2 , 2
+
2ueb d(b + 2)
3F4
1 _ b e+1 e + 1 . 1 4, 2 , 2 + 1 3 5+6 d+1 d + 1 2, 4 , 2 , 2 + 1
(3.1)
which is also believed to be a new result.
Corollary 3.2. In Theorem 2.1, if we take n =1, we get the following result:
S
1:1;0 2:0;0
[e : 1, 2] [d : 1, 2], [^ : 1,1] :
: [b : 1]; -
-2< 3F4
2
5—b e e+1 .
4 , 2, 2 ' 3 6+5 d d+1 2, 4 , 2, 2
- 3F4
3—b e e+1 .
4 , 2, 2 ' • _ u
3 b+3 d d+1 ; u 2, 4 , 2, 2
+
4uer( ^) d(b + 3)r( ^^)
4F5
4uer() d(b + 3)r( —)
1 7—b e+1 e + 1 .
1, 4 , 2 , 2 + 1
3 2 b+7 d+1 d + 1
2, 2, 4 , 2 , 2 +1
4F5
1 5 — b e+1 e + 1 .
1, 4 , 2 , 2 + 1
3 2 b+5 d+1 d + 1
2, 2, 4 , 2 , 2 +1
;u
,
which is also believed to be a new result.
(3.2)
u
2
u
x
2
u
x
2
u
2
u
2
u
u
2
u
2
2
u
;u
Corollary 3.3. In Theorem 2.2, if we take n = 1, we get the following result:
S
1:1;0 2:0;0
[e : 1, 2] : [b : 1];
[d :1,2], [^ : 1,1] : -;
2
3-5 e e+1. '
4 , 2, 2 ' ■ _„2 1 5+3 d d+1 ; u
2' 4 ' 2' 2 .
, 2ue(b - 1) F +--,,, . 3F4
+ 3F4
1-b e e+1. '
4 , 2, 2 ' ; _u2 1 b+1 d d+1 ; u
2' 4 ' 2' 2 .
d(b + 1) + 2ue F
+--r 3F4
d
5 — b e+1 e 1 1 .
4 , 2 , 2 + ;
3 b+5 d+1 d 1 1 ; 2' 4 ' 2 ' 2 + 1
+
3 — b e+1 e I 1 .
4 , 2 , 2 + 3 5+3 d+1 d 1 1 j 2' 4 ' 2 ' 2 + 1
;u
(3.3)
which is also believed to be a new result.
Similarly, other results can be obtained by giving values to n. However, we prefer to omit the details.
u
2
u
2
Concluding Remark
In this present note, we have provided certain new transformation formulas in the most general case for any n e N0 for the Srivastavas-Doust double hypergeometric functions with the help of certain general reduction formulas for Humbert's functions. The results are derived by employing the well-known Beta integral method. Interested reader can develop further new and interesting formulas by employing beta integral method.
Competing interests. The authors declare that they have no competing interests.
References
[1] T.Appell, K. de Feriet, Fonctions Hypergeometriques et Hyperspheriques; Polynomes d'Hermite, Gauthier-Villers, Paris, 1926.
[2] W.N.Bailey, Generalized Hypergeometric Series, Cambridge University Press, Cambridge, 1935; Reprinted by Stechert-Hafner, New York, 1964.
[3] Yu.A.Brychkov, Handbook of Special Functions, Derivatives, Integrals, Series and Other Formulas, Chapman & Hall/CRC Press, Boca Raton, 2008.
[4] Yu.A.Brychkov, Y.S.Kim, A.K.Rathie, On a new reduction formulas for the Humbert functions and $3, Integral Transforms Spec. Func., 28(2017), no. 5, 350-360, .
[5] J.L.Burchnall, T.W.Chaundy, Expansions of Appell's double hypergeometric functions, Quart. J. Math. (Oxford ser.), 11(1940), 249-270.
[6] J.L.Burchnall, T.W.Chaundy, Expansions of Appell's double hypergeometric functions (II), Quart. J. Math. (Oxford ser.), 12(1941), 112-128.
[7] A.Erdelyi, W.Magnus, F.Oberhettinger, F.C.Tricomi, Higher Transcendental Functions, Vol. I, McGraw-Hill Book Company, New York, Toronto and London, 1953.
[8] I.S.Gradshteyn, I.M.Ryzhyk, Tables of Integrals, Sreies and Products, Academic Press, New York, 2007.
[9] C.Krattenthaler, K.S.Rao, Automatic generation of hypergeometric identities by the beta integral method, J. Comput. Appl. Math., 160(2003), 159-173.
10] A.P.Prudnikov, Yu.A.Brychkov, O.I.Marichev, Integrals and Series, More Special Functions, Vol. 3, Gordon and Breach, New York, 1990.
11] E.D.Rainville,, Special Functions, Macmillan Company, New York, (1960); Reprinted by Chelsea Publishing Company, New York, 1971.
12] M.A.Rakha, A.K.Rathie, Generalizations of classical summation theorems for the series 2F1 and 3F2 with applications, Integral Transforms Spec. Func., 22(2011), no. 11, 823-840.
13] H.M.Srivastava, M.C.Daoust, Certain generalized Neumann expansions associated with Kampe de Feriet functions, Indag. Math., 31(1969) 449-457.
14] H.M.Srivastava, M.C.Daoust, On Eulerian integrals associated with Kampe de Feriet functions, Publ. Inst. Math. (Beograd) (N.S.), 9(1969), 199-202.
15] H.M.Srivastava, M.C.Daoust, A note on the convergence of Kampe de Feriet double hy-pergeometric series, Math. Nachr., 53(1972), 151-159.
16] H.M.Srivastava, P.W.Karlsson, Multiple Gaussian Hypergeometric Series, Halsted Press (Ellis Horwood Limited, Chichester), John Wiley and Sons, New York, Chichester, Brisbane and Toronto, 1985.
17] H.M. Srivastava, H.L.Manocha, A Treatise on Generating Functions, Halsted Press (Ellis Horwood Limited, Chichester), John Wiley and Sons, New York, Chichester, Brisbane and Toronto, 1984.
18] H.M.Srivastava, R.Panda, An integral representation for the product of two Jacobi polynomials, J. London Math. Soc., 12(1976), no. 2, 419-425.
19] C.Wei,, X.Wang, Y.Li, Certain transformations for multiple hypergeometric functions, Adv. Diff. Equ., 360(2013), 1-13. DOI: 10.1186/1687-1847-2013-360
Замечание о двух общих формулах приведения
и __1 и
для двойных гипергеометрических функции Шриваставы-Дауста
Мушарраф Али
Колледж
Шахджаханпур-242001, Уттар-Прадеш, Индия
Харш Вардхан Харш
Университет 1СЕА1 Джайпур-302031, Раджастхан, Индия
Арджун К. Рэти
Ведантский инженерно-технологический колледж Технический университет Раджастана Бунди-323021, Раджастхан, Индия
Аннотация. Цель этой заметки — предоставить две новые общие формулы приведения для двойных гипергеометрических функций Шриваставы-Дауста. Также приведено несколько интересных частных случаев.
Ключевые слова: гипергеометрические функции, двойные гипергеометрические функции Гум-берта, функции Аппеля, двойная гипергеометрическая функция Шриваставы и Дауста, метод бета-интеграла.
