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Combinatorial and hypergeometric identities via the Legendre polynomials––A computational approach
Applied Mathematics and Computation 166 (2005) 181–195 www.elsevier.com/locate/amc
Combinatorial and hypergeometric identities via the Legendre polynomials––A computational approach M.E.A. El-Mikkawy a
a,*
, G.-S. Cheon
b
Department of Mathematics, Faculty of Science, Mansoura University, Mansoura, 35516, Egypt b Department of Mathematics, Daejin University, Pocheon 487-711, S. Korea
Abstract In this paper old and new combinatorial and hypergeometric identities are obtained via the Legendre polynomials as one of the most commonly used orthogonal polynomials. The results are tested using MAPLE Computer Algebra System (CAS). Ó 2004 Elsevier Inc. All rights reserved. Keywords: Orthogonal polynomials; Legendre polynomials; Hypergeometric series; MAPLE
1. Introduction The area of orthogonal polynomials is a very active research area in mathematics as well as in applications in mathematical physics, engineering, and computer science (see, for instance, [1–10,13]).
*
Corresponding author. E-mail addresses: [email protected] (M.E.A. El-Mikkawy), [email protected] (G.-S. Cheon). 0096-3003/$ - see front matter Ó 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2004.04.066
182
M.E.A. El-Mikkawy, G.-S. Cheon / Appl. Math. Comput. 166 (2005) 181–195
It is well known that (see, for instance [11,12]) any orthogonal set of functions {/0, /1, . . ., /n} on the interval [a, b] with respect to the weight function w(x) P 0 satisfies (i) For each i, /i (x) has exact degree i. Z 0; i 6¼ j; (ii) b wðxÞ/i ðxÞ/j ðxÞdx ¼ hi > 0; i ¼ j: a
ð1:1Þ
Moreover, if {/0, /1, . . ., /n} is a set of orthogonal polynomials on [a, b], then the following properties are necessarily satisfied: (i) If g(x) is any polynomial of degree m 6 n, then g (x) can be expressed uniquely in the form m X gðxÞ ¼ ck /k ðxÞ ð1:2Þ k¼0
In particular, we have the matrix representation [14] 3 2 3 2 /0 ðxÞ 1 7 6 7 6 6x7 6 /1 ðxÞ 7 7 6 7 6 7 6 7 6 6 x2 7 6 /2 ðxÞ 7 7; 6 7 ¼ L6 7 6 7 6 6 .. 7 6 . 7 6 . 7 6 .. 7 5 4 5 4 xn
ð1:3Þ
/n ðxÞ
where L is a nonsingular lower triangular matrix of order n + 1. (ii) The orthogonal polynomials {/0, /1, . . ., /n} satisfies a three-term recurrence relation /kþ1 ðxÞ ¼ ðAk x þ Bk Þ/k ðxÞ C k /k1 ðxÞ; k ¼ 0; 1; 2; . . . n 1;
ð1:4Þ
where Ak, Bk, and Ck are real constants and does not depend on x and /1 (x) 0. The three-term recurrence (1.4) provides a way of generating orthogonal polynomials of a particular set. (iii) All zeros of the polynomial /j (x) are real, distinct, lie in the open interval (a, b) and the zeros of /j (x) and /j + 1 (x) separate each other. (iv) If g (x) is a polynomial of degree m 6 n 1, then g(x) is orthogonal to /n (x), that is Z b wðxÞgðxÞ/n ðxÞdx ¼ 0: ð1:5Þ a
M.E.A. El-Mikkawy, G.-S. Cheon / Appl. Math. Comput. 166 (2005) 181–195
183
Section 2 of the current paper is mainly devoted to study Legendre polynomials as one of the most common set of orthogonal polynomials. Most important orthogonal polynomials can be written as terminating hypergeometric. Therefore section 3 is mainly concerned with this type of series.
2. Legendre polynomials In this section we shall be concerned with one of the most common set of orthogonal polynomials. It is the set of the Legendre polynomials {P0 (x), P1(x), . . ., Pn(x)} which are orthogonal on [a, b] = [1, 1] with respect to the weight function w(x) 1. The Legendre polynomials Pn(x) satisfy the linear, second order, homogeneous differential equation (sometimes called Legendre differential equation) ð1 x2 Þy 00 ðxÞ 2xy 0 ðxÞ þ nðn þ 1ÞyðxÞ ¼ 0
ð2:1Þ
for 1 < x < 1, and n P 0 and is given by any of the following forms (see, for instance, [1,15]) P n ðxÞ ¼
1 dn 2 ðx 1Þn ; 2n ðn!Þ dxn
n ¼ 0; 1; 2; . . . ;
2 n 1 X k n k nk P n ðxÞ ¼ n ð1Þ ð1 xÞ ð1 þ xÞ ; 2 k¼0 k n 2 n 1 X P n ðxÞ ¼ n ðx þ 1Þk ðx 1Þnk ; 2 k¼0 k
P n ðxÞ ¼
n X k¼0
2
k
n n 1 k
k
ð2:2Þ
n ¼ 0; 1; 2; . . . ;
n ¼ 0; 1; 2; . . . ;
k
ð1 xÞ ;
n ¼ 0; 1; 2; . . . ;
ð2:3Þ
ð2:4Þ
ð2:5Þ
k n X n nþk 1 ð1 xÞk ; P n ðxÞ ¼ 2 k n k¼0
n ¼ 0; 1; 2; . . . ;
ð2:6Þ
½n2 2n 2k n2k 1 X k n ð1Þ ; P n ðxÞ ¼ n x 2 k¼0 k n
n ¼ 0; 1; 2; . . . ;
ð2:7Þ
where hni n ; n even; ¼ 2n1 2 ; n odd: 2
ð2:8Þ
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M.E.A. El-Mikkawy, G.-S. Cheon / Appl. Math. Comput. 166 (2005) 181–195
The first five Legendre polynomials are P0(x) = 1, P1(x) = x, P 2 ðxÞ ¼ 1 ð3x2 1Þ, P 3 ðxÞ ¼ 12 ð5x3 3xÞ, P 4 ðxÞ ¼ 18 ð35x4 30x2 þ 3Þ and they satisfy 2 the three-term recurrence relation P nþ1 ðxÞ ¼
1 ½ð2n þ 1ÞxP n ðxÞ nP n1 ðxÞ: nþ1
ð2:9Þ
Legendre polynomials satisfy the following interesting properties: P n ðxÞ is an oddðevenÞfunction when n is oddðevenÞ;
ð2:10Þ
P n ð1Þ ¼ 1;
ð2:11Þ
n ¼ 0; 1; 2; . . . ;
P n ðxÞ ¼ ð1Þn P n ðxÞ; 1
ð1 2xt þ t2 Þ2 ¼
1 X
n ¼ 1; 2; 3; . . . ; P n ðxÞtn ;
jtj 6 1;
ð2:12Þ ð2:13Þ
n¼0
Z
(
1
P n ðxÞP m ðxÞdx ¼
1
dn ð2nÞ! ; P n ðxÞ ¼ n 2 ðn!Þ dxn Z
1
1
f ðxÞP n ðxÞdx ¼
0;
m 6¼ n;
2 ; 2nþ1
m ¼ n;
ð2:14Þ
n ¼ 0; 1; 2; . . . ; 1 2n n!
Z
1
1
n
ð1 x2 Þ
ð2:15Þ dn f ðxÞ dx; dxn
ð2:16Þ
where f (x) is differentiable to the nth order on 1 6 x 6 1 (see [17]). For the sake of completeness it is convenient to state the following result xa whose proof may be obtained by using the substitution t ¼ ba together with the definition of the Beta function B(p, q) which is related to the Gamma function by Bðp; qÞ ¼ CðpÞCðqÞ . CðpþqÞ Theorem 2.1 Z b p1 q1 pþq1 pþq1 CðpÞCðqÞ : ðx aÞ ðb xÞ dx ¼ ðb aÞ Bðp; qÞ ¼ ðb aÞ Cðp þ qÞ a ð2:17Þ In particular, we have for a = 1 and b = 1 Z 1 CðpÞCðqÞ q1 p1 : ð1 xÞ ð1 þ xÞ dx ¼ 2pþq1 Bðp; qÞ ¼ 2pþq1 Cðp þ qÞ 1
ð2:18Þ
M.E.A. El-Mikkawy, G.-S. Cheon / Appl. Math. Comput. 166 (2005) 181–195
185
At this point it is convenient to add some more results obtained by these authors. The proofs are omitted here for the sake of space requirements. All results are tested using MAPLE.
P n ð0Þ ¼
Z
1
1
Z
1
1
Z
1
1
8 0 > > < ð1Þ > > :
n=2
n odd;
n
2n
ð2:19Þ
n 2
n even;
pffiffiffi P n ðxÞ 2 2 pffiffiffiffiffiffiffiffiffiffiffi dx ¼ ; 2n þ 1 1x
n ¼ 0; 1; 2; . . . ;
pffiffiffi P n ðxÞ n 2 2 pffiffiffiffiffiffiffiffiffiffiffi dx ¼ ð1Þ ; 2n þ 1 1þx 8 <0 P n ðxÞ 2 pffiffiffiffiffiffiffiffiffiffiffiffiffi dx ¼ C nþ1 : ðn 2 Þ 1 x2 ð Þ!
n ¼ 0; 1; 2; . . .
ð2:20Þ
ð2:21Þ
n odd; n even;
ð2:22Þ
2
Z
1
1
Z
xP n ðxÞ pffiffiffiffiffiffiffiffiffiffiffi2 dx ¼ 2 1x
Z
1
xP n ðxÞ pffiffiffiffiffiffiffiffiffiffiffi2 dx 1x C n2 C n2 þ 1 ¼ n 1 n 3 ; C 2þ2 C 2þ2 0
(
1 m
ðx þ aÞ P n ðxÞdx ¼
2
2 ðn!Þ ð2nþ1Þ!
m ¼ n;
;
ða is any real numberÞ;
Z 0
1
P n ðxÞdx ¼
8 > > <
1n1 4 2 > > : 0;
ð2:23Þ
m ¼ 0; 1; 2; . . . ; n 1;
0; nþ1
1
n odd;
n1
ð2:24Þ
n1 2 nþ1
; n ¼ 1; 3; 5; . . . ;
ð2:25Þ
n ¼ 2; 4; 6; . . .
Using (2.1)–(2.25), we obtained the following results: 8 0; n odd; !2 > > < n X n ! k n ð1Þ ¼ n=2 > n even; k > k¼0 : ð1Þ n 2
ð2:26Þ
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M.E.A. El-Mikkawy, G.-S. Cheon / Appl. Math. Comput. 166 (2005) 181–195
n X m ð1Þ X k¼0 r¼0
2 2 n m 0; m 6¼ n; k r ¼ nþm 1 m ¼ n; kþr
kþr
ð2:27Þ
2 n X 1 1 k n ð1Þ Cðn k þ 1ÞC k þ ¼C nþ ; 2 2 k k¼0
n ¼ 0; 1; 2; ð2:28Þ
8 > 0; ! 2 < n X n 1 1 k n nþ1 2 ð1Þ C nkþ C kþ ¼ > ; 2 2 k : n C 2 k¼0
n odd; n even;
2
ð2:29Þ
" #2 n=2 X 2n C nþ1 2n 2k C n2 k þ 12 k n n ¼ pffiffiffi n2 ð1Þ ; k n p ! C k þ 1 k¼0 2 2
n even; ð2:30Þ
½n2 X 2n 2k k n ð1Þ ¼ 2n ; k n k¼0 ½n2 X ð1Þk
ð2:31Þ
n 2n 2k k n ð2n 2k þ 1Þ
k¼0
n ¼ 0; 1; 2; . . . :
2
¼
ð2n n!Þ ; ð2n þ 1Þ!
n ¼ 0; 1; 2; 3; . . . ;
n1 2 X 2n 2k C n2 k þ 1 2n C n2 C n2 þ 1 k n ð1Þ ¼ pffiffiffi nþ1 ; p C 2 C nþ3 Cðn2 k þ 32Þ k n k¼0 2
ð2:32Þ
n odd; ð2:33Þ
2 n X 0; k n ð1Þ k!ðm þ n kÞ! ¼ ðn!Þ2 ; k k¼0
m ¼ 0; 1; 2; . . . ; n 1; m ¼ n; ð2:34Þ
n X k¼0
k
ð1Þ
2 n k
ðn kÞ!ðm þ kÞ! ¼
m ¼ 0; 1; 2; . . . ; n 1;
0; n
2
ð1Þ ðn!Þ ; m ¼ n;
ð2:35Þ
M.E.A. El-Mikkawy, G.-S. Cheon / Appl. Math. Comput. 166 (2005) 181–195
8 n 2n 2k n1 > ½ < ð1Þn1 2 X 2 nþ1 ; k n k n1 ¼ ð1Þ 2 > n 2k þ 1 : k¼0 0; n 2
187
n ¼ 1; 3; 5; . . . ; n ¼ 2; 4; 6; . . . ;
ð2:36Þ
n X k¼0
2k
n 1
2n k ¼ ; ðn kÞ!ðn þ k þ 1Þ! ð2n þ 1Þ!
n 1 k ð Þ 2 n 2 X 2n ðn!Þ k ¼ ; ðn þ k þ 1Þ ð2n þ 1Þ! k¼0
ð2:37Þ
n ¼ 0; 1; 2; . . . ;
ð2:38Þ
n ¼ 0; 1; 2; . . . ;
n 1 n X 1 k ; ¼ ðn kÞ!ðn þ k þ 1Þ! ð2n þ 1Þ! k¼0 n ðn!Þ2 k ¼ ; ðn þ k þ 1Þ ð2n þ 1Þ!
ð2:39Þ
n ¼ 0; 1; 2; . . . ;
k
n ð1Þ X k¼0
ð2:40Þ
n ¼ 0; 1; 2; . . . ;
n n 1
n X k¼0
k k ðn þ k þ 1Þ k
n ð1Þ X k¼0
P n ð3Þ ¼
2
n
¼
ð1Þ ðn!Þ ; ð2n þ 1Þ!
ð2:41Þ
n ¼ 0; 1; 2; . . . ;
n nþk
n X k¼0
2k
2 n k
2
n
k n ðn þ k þ 1Þ
¼
¼
ð1Þ ðn!Þ ; ð2n þ 1Þ!
n X n nþk k¼0
k
n
n ¼ 0; 1; 2; . . . ;
¼
ð2:42Þ
n X 2k nþk k¼0
k
2k ð2:43Þ
showing that Pn (3) is an odd positive integer for n = 0,1,2, . . . n nþk k ð1Þ n X k k ¼ 0; n ¼ 1; 2; 3; . . . ; kþ1 k¼0
ð2:44Þ
188
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k
n X
ð1Þ
k¼0
k
n X
ð1Þ
k¼0
n
!
nþk
!
k k 2k þ 1 n k
!
¼
nþk
1 ; 2n þ 1
n ¼ 0; 1; 2; 3; . . . ;
!
C k þ 12
k Cðk þ 1Þ
8 > < 0;
¼
> : p1ffiffip
Cðn2þ12Þ Cðn2þ1Þ
ð2:45Þ
n ¼ 1; 3; 5; . . .
2 ;
n ¼ 0; 2; 4; . . . ; ð2:46Þ
n X nþk k n n ð1Þ ¼ ð1Þ ; k n k¼0
k n X 1 2 k¼0
n k
!
nþk
!
n
¼
n ¼ 0; 1; 2; . . . ;
8 0; > > > <
n
n
ð1Þ2 > > > :
n 2 2n
ð2:47Þ
n odd;
!
ð2:48Þ ;
n even;
3. Hypergeometric and combinatorial identities Hypergeometric series are very important in Mathematics. Many of the familiar functions of analysis are hypergeometric. These include the exponential, logarithmic, binomial, trigonometric, and Bessel functions. Perhaps more interesting is the fact that orthogonal polynomials can be represented by hypergeometric functions. Definition 3.1 [1]. The generalized hypergeometric series is given by p F q ða1 ; . . . ; ap ; b1 ; . . . ; bq ; xÞ ¼
1 X
Ak x k ¼
k¼0
1 X ða1 Þk . . . ðap Þk xk ; ðb1 Þk . . . ðbq Þk k! k¼0
ð3:1Þ
Qk where ðaÞk ¼ j¼1 ða þ j 1Þ ¼ CðaþkÞ denotes the Pochhammer symbol or CðaÞ rising factorial. Ak is a hypergeometric term and fulfils the recurrence equation (k 2 N) Akþ1 ¼
ðk þ a1 Þ ðk þ ap Þ Ak ðk þ b1 Þ . . . :ðk þ bq Þðk þ 1Þ
with the initial value A0 ¼ 1:
M.E.A. El-Mikkawy, G.-S. Cheon / Appl. Math. Comput. 166 (2005) 181–195
189
Thus, a hypergeometric series is completely characterized by the two sequences a = (a1, . . ., ap), b = (b1, . . ., bq) and the argument x. The aiÕs and the biÕs are called respectively, the upper and the lower parameters of the series. The biÕs are not permitted to be non-positive otherwise we will have a division by 0. A negative ai on the other hand, turns the hypergeometric series in (3.1) into a finite series and hence a polynomial in x. In this paper we are only interested in the case where x as well as the parameters ai and bi are real. Some authors use the notation 0
a1 ; . . . ; ap
B
1 C ;xA
pF q@
b1 ; . . . ; bq instead of pFq(a1, . . ., ap;b1, . . ., bq;x) and F(a, b; c; x) instead of 2F1(a, b;c;x). In MAPLE the hypergeometric series is given as hypergeom (plist, qlist, x), where plist = [a1, . . ., ap] and qlist = [b1, . . ., bq]. Definition 3.2 [16]. A generalized hypergeometric Pq Pfunction p b1, . . ., bq;x) is said to be k-balanced if i¼1 bi ¼ k þ i¼1 ai .
pFq(a1, . . ., ap;
Definition 3.3 [16]. A generalized hypergeometric function pFq(a1, . . ., ap; b1, . . ., bq;x) isPsaid to be Saalschutzian if it is k-balanced with k = 1, P q p i¼1 bi ¼ 1 þ i¼1 ai . The Legendre polynomials Pn(x), n = 0,1,2,. . . can be represented by [1,15] 1x P n ðxÞ ¼ F n; n þ 1; 1; ; ð3:2Þ 2 P n ðxÞ ¼
1x 2
n 1þx F n; n; 1; ; 1x
2n x n n n1 1 1 ; n þ ; 2 ; F ; P n ðxÞ ¼ 2 2 2 2 x n n n1 1 n ; 1; 1 2 : P n ðxÞ ¼ x F ; 2 2 x
ð3:3Þ
ð3:4Þ
ð3:5Þ
Using sumtools library in MAPLE, we obtained the following results 2 X k n ð1Þ ¼ F ðn; n; 1; 1Þ; k k
ð3:6Þ
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M.E.A. El-Mikkawy, G.-S. Cheon / Appl. Math. Comput. 166 (2005) 181–195
X
ð1Þ
k
k
2 pffiffiffi n 1 1 ; n; 1; 1 ; Cðn k þ 1ÞC k þ ¼ ðn!Þ p F 2 2 k ð3:7Þ
X
ð1Þk
k
2 n 1 1 1 C n k þ Cðk þ 1Þ ¼ C n þ F n; n; n þ ; 1 ; 2 2 2 k ð3:8Þ
X 2n 2k 2n 1 1 1 1 k n ð1Þ ¼ F n; n þ ; n þ ; 1 ; 2 2 2 2 k n n k ð3:9Þ X n 2n 2k C n2 k þ 12 ð2nÞ! p F 12 n; 12 n; n þ 12 ; 1 k ð1Þ ; ¼ 3 Cðn2 k þ 1Þ k n 4n C n þ 1 C n þ 1 k 2
2
2
ð3:10Þ X 2n 2k C n2 k þ 1 n ð2nÞ! p F 12 n þ 12 ; 12 n 12 ; n þ 12 ; 1 k ð1Þ ; ¼ 3 C n2 k þ 32 n k 4n ðn þ 1Þ C n þ 1 C n þ 1 k 2
2
2
ð3:11Þ
2 X k n ð1Þ k!ðm þ n kÞ! ¼ ðm þ nÞ!F ðn; n; m n; 1Þ; k k
ð3:12Þ
2 X k n ð1Þ ðn kÞ!ðm þ kÞ! ¼ ðmÞ!ðnÞ!F ðn; m þ 1; 1; 1Þ; k k
ð3:13Þ
n 1 X 2 F ðn; n þ 1; n þ 2; 12Þ k ; ¼ ðnÞ!ðn þ 1Þ! ðn kÞ!ðn þ k þ 1Þ! k
ð3:14Þ
n X F n; n þ 1; n þ 2; 12 k ¼ ; nþkþ1 nþ1 k
ð3:15Þ
k
k 12
n 1 X F ðn; n þ 1; n þ 2; 1Þ k ; ¼ ðnÞ!ðn þ 1Þ! ðn kÞ!ðn þ k þ 1Þ! k
ð3:16Þ
M.E.A. El-Mikkawy, G.-S. Cheon / Appl. Math. Comput. 166 (2005) 181–195
k
X k
ð1Þ
n
!
2
X k
k
!
n 1
!
6 6
ð1Þ
n k
k
!
3
n; n þ 1; n þ 1
7 ;17 5
3F 24
k nþkþ1 k
X
ð3:17Þ
k F ðn; n þ 1; n þ 2; 1Þ ¼ ; nþkþ1 nþ1
n
191
n þ 2; 1 nþ1
¼ !
nþk
n Cðk þ 1Þ
; 21
C k þ 12 ¼
ð3:18Þ
6 pffiffiffi p 3F 26 4
; n; n þ 1
2
3 7 ;17 5;
1; 1 ð3:19Þ
X ð1Þk k
n
k
k
n X ð1Þk k
X 1k 2 k
nþk
k
! ¼ F ðn; n þ 1; 1; 1Þ;
n
k n
X ð1Þk
!
!
nþk
!
n kþ1 !
nþk
!
k
ð3:21Þ
¼ F ðn; n þ 1; 2; 1Þ; !
21 6 ¼ 3F 26 4
n 2k þ 1
n
ð3:20Þ
nþk n
!
2
3
; n; n þ 1
7 ;17 5;
3 ;1 2
ð3:22Þ
1 ¼ F n; n þ 1; 1; : 2
ð3:23Þ
Using (3.6)–(3.23) together with the results presented in section 2 yields the following hypergeometric and combinatorial identities
F ðn; n; 1; 1Þ ¼
X k
ð1Þk
n k
!2 ¼
8 > > <
0; n
2 > > : ð1Þ
n n 2
!
n odd ; n even;
ð3:24Þ
192
M.E.A. El-Mikkawy, G.-S. Cheon / Appl. Math. Comput. 166 (2005) 181–195
1 ; n; 1; 1 F 2
!2 n 1 X 1 k ¼ pffiffiffi ð1Þ Cðn k þ 1ÞC k þ 2 n! p k k C n þ 12 pffiffiffi ; n ¼ 0; 1; 2; . . . ; ¼ n! p
ð3:25Þ
1 F n;n;n þ ;1 2 n X 1 k ¼ ð1Þ 1 C nþ2 k k
!2 1 n C n k þ Cðk þ 1Þ ¼ ð1Þ ; n ¼ 0;1;2;...; 2 ð3:26Þ
1 1 1 1 F n; n þ ; n þ ; 1 2 2 2 2 ! ! n 2n 2k X 1 k ! ¼ ð1Þ ¼ 2n k n k
2n !; 2n
n
ð3:27Þ
n
k-balanced with k ¼ 0; n ¼ 0; 1; 2; . . . ;
F
12 n; 12 n;n þ 12 ;1
¼
¼
4n Cðn2þ12Þ½Cðn2þ1Þ
3
ð2nÞ!p
P
8 <0
k ð1Þ
n k
!
2n 2k
!
n
Cðn2kþ12Þ Cðn2kþ1Þ
n odd;
: 8 Cð n
k
Þ½Cpðffiffi ð2nÞ!p p
nþ1 2
nþ1 2
Þ
;
3
n even; ð3:28Þ
1 1 1 1 1 F n þ ; n ; n þ ; 1 2 2 2 2 2 n n 1 3 n 2n 2k 4 ðn þ 1Þ C 2 þ 1 C 2 þ 2 X k n ¼ ð1Þ k ð2nÞ! p k n nþ1 n 3 n 3n2 2 C kþ1 2 nC 2 C 2 ¼ pffiffiffi 2n ; n ¼ 1; 3; 5; . . . ; 3 ð2nÞ!p p C 2kþ2 ð3:29Þ
M.E.A. El-Mikkawy, G.-S. Cheon / Appl. Math. Comput. 166 (2005) 181–195
n F ðn; n; 2n; 1Þ ¼
X k ð1Þ k
193
!2
k ! 2n k
¼
1 !; 2n
k-balanced with k ¼ 0; n ¼ 0; 1; 2; . . . ;
ð3:30Þ
n
F ðn; n þ 1; 1; 1Þ ¼
X k
n
¼ ð1Þ ;
n nþk ð1Þ k n k
k-balanced with k ¼ 0; n ¼ 0; 1; 2; . . . ;
k
n 1
ð3:31Þ
X 2 1 k F n; n þ 1; n þ 2; ¼ ðnÞ!ðn þ 1Þ! 2 ðn kÞ!ðn þ k þ 1Þ! k 1k n X 2 2n k ¼ ðn þ 1Þ ¼ ; n ¼ 0; 1; 2; . . . ; 2n þ 1 n þ k þ 1 k n
n 1
P
k ðn kÞ!ðn þ k þ 1Þ! k k n X ð1Þ k 1 ¼ ðn þ 1Þ ; n ¼ 0; 1; 2; . . . ; ¼ 2n þ 1 nþkþ1 k n
F ðn; n þ 1; n þ 2; 1Þ ¼ ðnÞ!ðn þ 1Þ!
2 6
n; n þ 1; n þ 1
3F 24
n þ 2; 1 n ð1Þ ; ¼ 2n þ 1 n
ð3:32Þ
ð3:33Þ
n n 1
3
X 7 ; 1 5 ¼ ðn þ 1Þ k
k
k nþkþ1
k-balanced with k ¼ 0; n ¼ 0; 1; 2; . . . ;
ð3:34Þ
194
M.E.A. El-Mikkawy, G.-S. Cheon / Appl. Math. Comput. 166 (2005) 181–195
21 6 6
; n; n þ 1 2
3F 24
1; 1 8 0; > < 2 ¼ Cðnþ1 2 Þ > 1 :p Cðn2þ1Þ
3 7 1 X ;17 5 ¼ pffiffiffi p k
ð1Þ
n
k
k
!
nþk
n Cðk þ 1Þ
!
C k þ 12
n ¼ 1; 3; 5; . . . ; ð3:35Þ
: n ¼ 0; 2; 4; . . . ;
n nþk X k n k F ðn; n þ 1; 2; 1Þ ¼ ¼0 ð1Þ k þ 1 k
ð3:36Þ
Saalschutzian type; n ¼ 1; 2; 3; . . . ; 21 6
3F 24
; n; n þ 1 2 3 ;1 2
n nþk
3
7 X k ;15 ¼ ð1Þ k
k
n 2k þ 1
¼
1 ; 2n þ 1
ð3:37Þ
k-balanced with k ¼ 0; n ¼ 0; 1; 2; . . . ; X k n nþk 1 1 F n; n þ 1; 1; ¼ 2 2 k n k 8 0; n odd > > < n n k-balanced with k ¼ 0; n ¼ 0; 1; 2; . . . ; ¼ ð1Þ2 n > > 2 : ; n even; 2n ð3:38Þ All results are tested using MAPLE.
References [1] W. Koepf, Software for the algorithmic work with orthogonal polynomials and special functions, ETNA, Electron. Trans. Numer. Anal. 9 (1999) 77–101. [2] M.E.H. Ismail, More on electrostatic models for zeros of orthogonal polynomials, Numer. Funct. Anal. Optim. 21 (1–2) (2000) 191–204. [3] D. Zilin, M. Eleftheriou, J.E. Moreira, C. Yap, Hypergeometric functions in exact geometric computation, Electron. Notes in Theoret. Comput. Science 66 (1) (2002).
M.E.A. El-Mikkawy, G.-S. Cheon / Appl. Math. Comput. 166 (2005) 181–195
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[4] P. Coolen-Schrijner, E.A. vanDoorn, Analysis of random walks using orthogonal polynomials, J. Comput. Appl. Math. 99 (1–2) (1998) 387–399. [5] P. Fischer, J. Prestin, Wavelets based on orthogonal polynomials, Math. Comp. 66 (1997) 1593–1618. [6] R. Askey, Finite differences and orthogonal polynomials, Rendiconti di Matematica, Series V II, vol. 4, Rome, 1994, pp. 135–144. [7] W. Gautschi, Orthogonal polynomials and quadrature, ETNA, Electron. Trans. Numer. Anal. 9 (1999) 65–76. [8] O. Renault, A new algorithm for computing orthogonal polynomials, J. Comput. Appl. Math. 75 (2) (1996) 231–248. [9] G.E. Andrew, R. Askey, R. Roy, Special Functions, Cambridge University, Cambridge, 1999. [10] A.D. Poularikas, Legendre polynomials, The Handbook of Formulas and Tables for Signal Processing, CRC Press LLC, Boca Raton, 1999. [11] R.L. Burden, J.D. Faires, Numerical Analysis, 7th ed., Brooks & Cole Publishing, 2001. [12] S.D. Conte, C. de Boor, Elementary Numerical Analysis: An Algorithmic Approach, 3rd ed., McGraw Inc., New York, 1980. [13] V.S. Adamchick, A certain series associated with CatalanÕs constant, Z. Anal. Anwend 21 (3) (2002) 817–826. [14] G.S. Cheon, M.E.A. El-Mikkawy, Matrix representation for orthogonal polynomials (in preparation). [15] G.S. Cheon, M.E.A. El-Mikkawy, Symmetric extended Pascal matrices via hypergeometric functions, Appl. Math. Computat. (in press). [16] Available from http://mathworld.wolfram.com. [17] Available from http://www.efunda.com/math/Legendre.
Combinatorial and hypergeometric identities via the Legendre polynomials––A computational approach M.E.A. El-Mikkawy a
a,*
, G.-S. Cheon
b
Department of Mathematics, Faculty of Science, Mansoura University, Mansoura, 35516, Egypt b Department of Mathematics, Daejin University, Pocheon 487-711, S. Korea
Abstract In this paper old and new combinatorial and hypergeometric identities are obtained via the Legendre polynomials as one of the most commonly used orthogonal polynomials. The results are tested using MAPLE Computer Algebra System (CAS). Ó 2004 Elsevier Inc. All rights reserved. Keywords: Orthogonal polynomials; Legendre polynomials; Hypergeometric series; MAPLE
1. Introduction The area of orthogonal polynomials is a very active research area in mathematics as well as in applications in mathematical physics, engineering, and computer science (see, for instance, [1–10,13]).
*
Corresponding author. E-mail addresses: [email protected] (M.E.A. El-Mikkawy), [email protected] (G.-S. Cheon). 0096-3003/$ - see front matter Ó 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2004.04.066
182
M.E.A. El-Mikkawy, G.-S. Cheon / Appl. Math. Comput. 166 (2005) 181–195
It is well known that (see, for instance [11,12]) any orthogonal set of functions {/0, /1, . . ., /n} on the interval [a, b] with respect to the weight function w(x) P 0 satisfies (i) For each i, /i (x) has exact degree i. Z 0; i 6¼ j; (ii) b wðxÞ/i ðxÞ/j ðxÞdx ¼ hi > 0; i ¼ j: a
ð1:1Þ
Moreover, if {/0, /1, . . ., /n} is a set of orthogonal polynomials on [a, b], then the following properties are necessarily satisfied: (i) If g(x) is any polynomial of degree m 6 n, then g (x) can be expressed uniquely in the form m X gðxÞ ¼ ck /k ðxÞ ð1:2Þ k¼0
In particular, we have the matrix representation [14] 3 2 3 2 /0 ðxÞ 1 7 6 7 6 6x7 6 /1 ðxÞ 7 7 6 7 6 7 6 7 6 6 x2 7 6 /2 ðxÞ 7 7; 6 7 ¼ L6 7 6 7 6 6 .. 7 6 . 7 6 . 7 6 .. 7 5 4 5 4 xn
ð1:3Þ
/n ðxÞ
where L is a nonsingular lower triangular matrix of order n + 1. (ii) The orthogonal polynomials {/0, /1, . . ., /n} satisfies a three-term recurrence relation /kþ1 ðxÞ ¼ ðAk x þ Bk Þ/k ðxÞ C k /k1 ðxÞ; k ¼ 0; 1; 2; . . . n 1;
ð1:4Þ
where Ak, Bk, and Ck are real constants and does not depend on x and /1 (x) 0. The three-term recurrence (1.4) provides a way of generating orthogonal polynomials of a particular set. (iii) All zeros of the polynomial /j (x) are real, distinct, lie in the open interval (a, b) and the zeros of /j (x) and /j + 1 (x) separate each other. (iv) If g (x) is a polynomial of degree m 6 n 1, then g(x) is orthogonal to /n (x), that is Z b wðxÞgðxÞ/n ðxÞdx ¼ 0: ð1:5Þ a
M.E.A. El-Mikkawy, G.-S. Cheon / Appl. Math. Comput. 166 (2005) 181–195
183
Section 2 of the current paper is mainly devoted to study Legendre polynomials as one of the most common set of orthogonal polynomials. Most important orthogonal polynomials can be written as terminating hypergeometric. Therefore section 3 is mainly concerned with this type of series.
2. Legendre polynomials In this section we shall be concerned with one of the most common set of orthogonal polynomials. It is the set of the Legendre polynomials {P0 (x), P1(x), . . ., Pn(x)} which are orthogonal on [a, b] = [1, 1] with respect to the weight function w(x) 1. The Legendre polynomials Pn(x) satisfy the linear, second order, homogeneous differential equation (sometimes called Legendre differential equation) ð1 x2 Þy 00 ðxÞ 2xy 0 ðxÞ þ nðn þ 1ÞyðxÞ ¼ 0
ð2:1Þ
for 1 < x < 1, and n P 0 and is given by any of the following forms (see, for instance, [1,15]) P n ðxÞ ¼
1 dn 2 ðx 1Þn ; 2n ðn!Þ dxn
n ¼ 0; 1; 2; . . . ;
2 n 1 X k n k nk P n ðxÞ ¼ n ð1Þ ð1 xÞ ð1 þ xÞ ; 2 k¼0 k n 2 n 1 X P n ðxÞ ¼ n ðx þ 1Þk ðx 1Þnk ; 2 k¼0 k
P n ðxÞ ¼
n X k¼0
2
k
n n 1 k
k
ð2:2Þ
n ¼ 0; 1; 2; . . . ;
n ¼ 0; 1; 2; . . . ;
k
ð1 xÞ ;
n ¼ 0; 1; 2; . . . ;
ð2:3Þ
ð2:4Þ
ð2:5Þ
k n X n nþk 1 ð1 xÞk ; P n ðxÞ ¼ 2 k n k¼0
n ¼ 0; 1; 2; . . . ;
ð2:6Þ
½n2 2n 2k n2k 1 X k n ð1Þ ; P n ðxÞ ¼ n x 2 k¼0 k n
n ¼ 0; 1; 2; . . . ;
ð2:7Þ
where hni n ; n even; ¼ 2n1 2 ; n odd: 2
ð2:8Þ
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M.E.A. El-Mikkawy, G.-S. Cheon / Appl. Math. Comput. 166 (2005) 181–195
The first five Legendre polynomials are P0(x) = 1, P1(x) = x, P 2 ðxÞ ¼ 1 ð3x2 1Þ, P 3 ðxÞ ¼ 12 ð5x3 3xÞ, P 4 ðxÞ ¼ 18 ð35x4 30x2 þ 3Þ and they satisfy 2 the three-term recurrence relation P nþ1 ðxÞ ¼
1 ½ð2n þ 1ÞxP n ðxÞ nP n1 ðxÞ: nþ1
ð2:9Þ
Legendre polynomials satisfy the following interesting properties: P n ðxÞ is an oddðevenÞfunction when n is oddðevenÞ;
ð2:10Þ
P n ð1Þ ¼ 1;
ð2:11Þ
n ¼ 0; 1; 2; . . . ;
P n ðxÞ ¼ ð1Þn P n ðxÞ; 1
ð1 2xt þ t2 Þ2 ¼
1 X
n ¼ 1; 2; 3; . . . ; P n ðxÞtn ;
jtj 6 1;
ð2:12Þ ð2:13Þ
n¼0
Z
(
1
P n ðxÞP m ðxÞdx ¼
1
dn ð2nÞ! ; P n ðxÞ ¼ n 2 ðn!Þ dxn Z
1
1
f ðxÞP n ðxÞdx ¼
0;
m 6¼ n;
2 ; 2nþ1
m ¼ n;
ð2:14Þ
n ¼ 0; 1; 2; . . . ; 1 2n n!
Z
1
1
n
ð1 x2 Þ
ð2:15Þ dn f ðxÞ dx; dxn
ð2:16Þ
where f (x) is differentiable to the nth order on 1 6 x 6 1 (see [17]). For the sake of completeness it is convenient to state the following result xa whose proof may be obtained by using the substitution t ¼ ba together with the definition of the Beta function B(p, q) which is related to the Gamma function by Bðp; qÞ ¼ CðpÞCðqÞ . CðpþqÞ Theorem 2.1 Z b p1 q1 pþq1 pþq1 CðpÞCðqÞ : ðx aÞ ðb xÞ dx ¼ ðb aÞ Bðp; qÞ ¼ ðb aÞ Cðp þ qÞ a ð2:17Þ In particular, we have for a = 1 and b = 1 Z 1 CðpÞCðqÞ q1 p1 : ð1 xÞ ð1 þ xÞ dx ¼ 2pþq1 Bðp; qÞ ¼ 2pþq1 Cðp þ qÞ 1
ð2:18Þ
M.E.A. El-Mikkawy, G.-S. Cheon / Appl. Math. Comput. 166 (2005) 181–195
185
At this point it is convenient to add some more results obtained by these authors. The proofs are omitted here for the sake of space requirements. All results are tested using MAPLE.
P n ð0Þ ¼
Z
1
1
Z
1
1
Z
1
1
8 0 > > < ð1Þ > > :
n=2
n odd;
n
2n
ð2:19Þ
n 2
n even;
pffiffiffi P n ðxÞ 2 2 pffiffiffiffiffiffiffiffiffiffiffi dx ¼ ; 2n þ 1 1x
n ¼ 0; 1; 2; . . . ;
pffiffiffi P n ðxÞ n 2 2 pffiffiffiffiffiffiffiffiffiffiffi dx ¼ ð1Þ ; 2n þ 1 1þx 8 <0 P n ðxÞ 2 pffiffiffiffiffiffiffiffiffiffiffiffiffi dx ¼ C nþ1 : ðn 2 Þ 1 x2 ð Þ!
n ¼ 0; 1; 2; . . .
ð2:20Þ
ð2:21Þ
n odd; n even;
ð2:22Þ
2
Z
1
1
Z
xP n ðxÞ pffiffiffiffiffiffiffiffiffiffiffi2 dx ¼ 2 1x
Z
1
xP n ðxÞ pffiffiffiffiffiffiffiffiffiffiffi2 dx 1x C n2 C n2 þ 1 ¼ n 1 n 3 ; C 2þ2 C 2þ2 0
(
1 m
ðx þ aÞ P n ðxÞdx ¼
2
2 ðn!Þ ð2nþ1Þ!
m ¼ n;
;
ða is any real numberÞ;
Z 0
1
P n ðxÞdx ¼
8 > > <
1n1 4 2 > > : 0;
ð2:23Þ
m ¼ 0; 1; 2; . . . ; n 1;
0; nþ1
1
n odd;
n1
ð2:24Þ
n1 2 nþ1
; n ¼ 1; 3; 5; . . . ;
ð2:25Þ
n ¼ 2; 4; 6; . . .
Using (2.1)–(2.25), we obtained the following results: 8 0; n odd; !2 > > < n X n ! k n ð1Þ ¼ n=2 > n even; k > k¼0 : ð1Þ n 2
ð2:26Þ
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M.E.A. El-Mikkawy, G.-S. Cheon / Appl. Math. Comput. 166 (2005) 181–195
n X m ð1Þ X k¼0 r¼0
2 2 n m 0; m 6¼ n; k r ¼ nþm 1 m ¼ n; kþr
kþr
ð2:27Þ
2 n X 1 1 k n ð1Þ Cðn k þ 1ÞC k þ ¼C nþ ; 2 2 k k¼0
n ¼ 0; 1; 2; ð2:28Þ
8 > 0; ! 2 < n X n 1 1 k n nþ1 2 ð1Þ C nkþ C kþ ¼ > ; 2 2 k : n C 2 k¼0
n odd; n even;
2
ð2:29Þ
" #2 n=2 X 2n C nþ1 2n 2k C n2 k þ 12 k n n ¼ pffiffiffi n2 ð1Þ ; k n p ! C k þ 1 k¼0 2 2
n even; ð2:30Þ
½n2 X 2n 2k k n ð1Þ ¼ 2n ; k n k¼0 ½n2 X ð1Þk
ð2:31Þ
n 2n 2k k n ð2n 2k þ 1Þ
k¼0
n ¼ 0; 1; 2; . . . :
2
¼
ð2n n!Þ ; ð2n þ 1Þ!
n ¼ 0; 1; 2; 3; . . . ;
n1 2 X 2n 2k C n2 k þ 1 2n C n2 C n2 þ 1 k n ð1Þ ¼ pffiffiffi nþ1 ; p C 2 C nþ3 Cðn2 k þ 32Þ k n k¼0 2
ð2:32Þ
n odd; ð2:33Þ
2 n X 0; k n ð1Þ k!ðm þ n kÞ! ¼ ðn!Þ2 ; k k¼0
m ¼ 0; 1; 2; . . . ; n 1; m ¼ n; ð2:34Þ
n X k¼0
k
ð1Þ
2 n k
ðn kÞ!ðm þ kÞ! ¼
m ¼ 0; 1; 2; . . . ; n 1;
0; n
2
ð1Þ ðn!Þ ; m ¼ n;
ð2:35Þ
M.E.A. El-Mikkawy, G.-S. Cheon / Appl. Math. Comput. 166 (2005) 181–195
8 n 2n 2k n1 > ½ < ð1Þn1 2 X 2 nþ1 ; k n k n1 ¼ ð1Þ 2 > n 2k þ 1 : k¼0 0; n 2
187
n ¼ 1; 3; 5; . . . ; n ¼ 2; 4; 6; . . . ;
ð2:36Þ
n X k¼0
2k
n 1
2n k ¼ ; ðn kÞ!ðn þ k þ 1Þ! ð2n þ 1Þ!
n 1 k ð Þ 2 n 2 X 2n ðn!Þ k ¼ ; ðn þ k þ 1Þ ð2n þ 1Þ! k¼0
ð2:37Þ
n ¼ 0; 1; 2; . . . ;
ð2:38Þ
n ¼ 0; 1; 2; . . . ;
n 1 n X 1 k ; ¼ ðn kÞ!ðn þ k þ 1Þ! ð2n þ 1Þ! k¼0 n ðn!Þ2 k ¼ ; ðn þ k þ 1Þ ð2n þ 1Þ!
ð2:39Þ
n ¼ 0; 1; 2; . . . ;
k
n ð1Þ X k¼0
ð2:40Þ
n ¼ 0; 1; 2; . . . ;
n n 1
n X k¼0
k k ðn þ k þ 1Þ k
n ð1Þ X k¼0
P n ð3Þ ¼
2
n
¼
ð1Þ ðn!Þ ; ð2n þ 1Þ!
ð2:41Þ
n ¼ 0; 1; 2; . . . ;
n nþk
n X k¼0
2k
2 n k
2
n
k n ðn þ k þ 1Þ
¼
¼
ð1Þ ðn!Þ ; ð2n þ 1Þ!
n X n nþk k¼0
k
n
n ¼ 0; 1; 2; . . . ;
¼
ð2:42Þ
n X 2k nþk k¼0
k
2k ð2:43Þ
showing that Pn (3) is an odd positive integer for n = 0,1,2, . . . n nþk k ð1Þ n X k k ¼ 0; n ¼ 1; 2; 3; . . . ; kþ1 k¼0
ð2:44Þ
188
M.E.A. El-Mikkawy, G.-S. Cheon / Appl. Math. Comput. 166 (2005) 181–195
k
n X
ð1Þ
k¼0
k
n X
ð1Þ
k¼0
n
!
nþk
!
k k 2k þ 1 n k
!
¼
nþk
1 ; 2n þ 1
n ¼ 0; 1; 2; 3; . . . ;
!
C k þ 12
k Cðk þ 1Þ
8 > < 0;
¼
> : p1ffiffip
Cðn2þ12Þ Cðn2þ1Þ
ð2:45Þ
n ¼ 1; 3; 5; . . .
2 ;
n ¼ 0; 2; 4; . . . ; ð2:46Þ
n X nþk k n n ð1Þ ¼ ð1Þ ; k n k¼0
k n X 1 2 k¼0
n k
!
nþk
!
n
¼
n ¼ 0; 1; 2; . . . ;
8 0; > > > <
n
n
ð1Þ2 > > > :
n 2 2n
ð2:47Þ
n odd;
!
ð2:48Þ ;
n even;
3. Hypergeometric and combinatorial identities Hypergeometric series are very important in Mathematics. Many of the familiar functions of analysis are hypergeometric. These include the exponential, logarithmic, binomial, trigonometric, and Bessel functions. Perhaps more interesting is the fact that orthogonal polynomials can be represented by hypergeometric functions. Definition 3.1 [1]. The generalized hypergeometric series is given by p F q ða1 ; . . . ; ap ; b1 ; . . . ; bq ; xÞ ¼
1 X
Ak x k ¼
k¼0
1 X ða1 Þk . . . ðap Þk xk ; ðb1 Þk . . . ðbq Þk k! k¼0
ð3:1Þ
Qk where ðaÞk ¼ j¼1 ða þ j 1Þ ¼ CðaþkÞ denotes the Pochhammer symbol or CðaÞ rising factorial. Ak is a hypergeometric term and fulfils the recurrence equation (k 2 N) Akþ1 ¼
ðk þ a1 Þ ðk þ ap Þ Ak ðk þ b1 Þ . . . :ðk þ bq Þðk þ 1Þ
with the initial value A0 ¼ 1:
M.E.A. El-Mikkawy, G.-S. Cheon / Appl. Math. Comput. 166 (2005) 181–195
189
Thus, a hypergeometric series is completely characterized by the two sequences a = (a1, . . ., ap), b = (b1, . . ., bq) and the argument x. The aiÕs and the biÕs are called respectively, the upper and the lower parameters of the series. The biÕs are not permitted to be non-positive otherwise we will have a division by 0. A negative ai on the other hand, turns the hypergeometric series in (3.1) into a finite series and hence a polynomial in x. In this paper we are only interested in the case where x as well as the parameters ai and bi are real. Some authors use the notation 0
a1 ; . . . ; ap
B
1 C ;xA
pF q@
b1 ; . . . ; bq instead of pFq(a1, . . ., ap;b1, . . ., bq;x) and F(a, b; c; x) instead of 2F1(a, b;c;x). In MAPLE the hypergeometric series is given as hypergeom (plist, qlist, x), where plist = [a1, . . ., ap] and qlist = [b1, . . ., bq]. Definition 3.2 [16]. A generalized hypergeometric Pq Pfunction p b1, . . ., bq;x) is said to be k-balanced if i¼1 bi ¼ k þ i¼1 ai .
pFq(a1, . . ., ap;
Definition 3.3 [16]. A generalized hypergeometric function pFq(a1, . . ., ap; b1, . . ., bq;x) isPsaid to be Saalschutzian if it is k-balanced with k = 1, P q p i¼1 bi ¼ 1 þ i¼1 ai . The Legendre polynomials Pn(x), n = 0,1,2,. . . can be represented by [1,15] 1x P n ðxÞ ¼ F n; n þ 1; 1; ; ð3:2Þ 2 P n ðxÞ ¼
1x 2
n 1þx F n; n; 1; ; 1x
2n x n n n1 1 1 ; n þ ; 2 ; F ; P n ðxÞ ¼ 2 2 2 2 x n n n1 1 n ; 1; 1 2 : P n ðxÞ ¼ x F ; 2 2 x
ð3:3Þ
ð3:4Þ
ð3:5Þ
Using sumtools library in MAPLE, we obtained the following results 2 X k n ð1Þ ¼ F ðn; n; 1; 1Þ; k k
ð3:6Þ
190
M.E.A. El-Mikkawy, G.-S. Cheon / Appl. Math. Comput. 166 (2005) 181–195
X
ð1Þ
k
k
2 pffiffiffi n 1 1 ; n; 1; 1 ; Cðn k þ 1ÞC k þ ¼ ðn!Þ p F 2 2 k ð3:7Þ
X
ð1Þk
k
2 n 1 1 1 C n k þ Cðk þ 1Þ ¼ C n þ F n; n; n þ ; 1 ; 2 2 2 k ð3:8Þ
X 2n 2k 2n 1 1 1 1 k n ð1Þ ¼ F n; n þ ; n þ ; 1 ; 2 2 2 2 k n n k ð3:9Þ X n 2n 2k C n2 k þ 12 ð2nÞ! p F 12 n; 12 n; n þ 12 ; 1 k ð1Þ ; ¼ 3 Cðn2 k þ 1Þ k n 4n C n þ 1 C n þ 1 k 2
2
2
ð3:10Þ X 2n 2k C n2 k þ 1 n ð2nÞ! p F 12 n þ 12 ; 12 n 12 ; n þ 12 ; 1 k ð1Þ ; ¼ 3 C n2 k þ 32 n k 4n ðn þ 1Þ C n þ 1 C n þ 1 k 2
2
2
ð3:11Þ
2 X k n ð1Þ k!ðm þ n kÞ! ¼ ðm þ nÞ!F ðn; n; m n; 1Þ; k k
ð3:12Þ
2 X k n ð1Þ ðn kÞ!ðm þ kÞ! ¼ ðmÞ!ðnÞ!F ðn; m þ 1; 1; 1Þ; k k
ð3:13Þ
n 1 X 2 F ðn; n þ 1; n þ 2; 12Þ k ; ¼ ðnÞ!ðn þ 1Þ! ðn kÞ!ðn þ k þ 1Þ! k
ð3:14Þ
n X F n; n þ 1; n þ 2; 12 k ¼ ; nþkþ1 nþ1 k
ð3:15Þ
k
k 12
n 1 X F ðn; n þ 1; n þ 2; 1Þ k ; ¼ ðnÞ!ðn þ 1Þ! ðn kÞ!ðn þ k þ 1Þ! k
ð3:16Þ
M.E.A. El-Mikkawy, G.-S. Cheon / Appl. Math. Comput. 166 (2005) 181–195
k
X k
ð1Þ
n
!
2
X k
k
!
n 1
!
6 6
ð1Þ
n k
k
!
3
n; n þ 1; n þ 1
7 ;17 5
3F 24
k nþkþ1 k
X
ð3:17Þ
k F ðn; n þ 1; n þ 2; 1Þ ¼ ; nþkþ1 nþ1
n
191
n þ 2; 1 nþ1
¼ !
nþk
n Cðk þ 1Þ
; 21
C k þ 12 ¼
ð3:18Þ
6 pffiffiffi p 3F 26 4
; n; n þ 1
2
3 7 ;17 5;
1; 1 ð3:19Þ
X ð1Þk k
n
k
k
n X ð1Þk k
X 1k 2 k
nþk
k
! ¼ F ðn; n þ 1; 1; 1Þ;
n
k n
X ð1Þk
!
!
nþk
!
n kþ1 !
nþk
!
k
ð3:21Þ
¼ F ðn; n þ 1; 2; 1Þ; !
21 6 ¼ 3F 26 4
n 2k þ 1
n
ð3:20Þ
nþk n
!
2
3
; n; n þ 1
7 ;17 5;
3 ;1 2
ð3:22Þ
1 ¼ F n; n þ 1; 1; : 2
ð3:23Þ
Using (3.6)–(3.23) together with the results presented in section 2 yields the following hypergeometric and combinatorial identities
F ðn; n; 1; 1Þ ¼
X k
ð1Þk
n k
!2 ¼
8 > > <
0; n
2 > > : ð1Þ
n n 2
!
n odd ; n even;
ð3:24Þ
192
M.E.A. El-Mikkawy, G.-S. Cheon / Appl. Math. Comput. 166 (2005) 181–195
1 ; n; 1; 1 F 2
!2 n 1 X 1 k ¼ pffiffiffi ð1Þ Cðn k þ 1ÞC k þ 2 n! p k k C n þ 12 pffiffiffi ; n ¼ 0; 1; 2; . . . ; ¼ n! p
ð3:25Þ
1 F n;n;n þ ;1 2 n X 1 k ¼ ð1Þ 1 C nþ2 k k
!2 1 n C n k þ Cðk þ 1Þ ¼ ð1Þ ; n ¼ 0;1;2;...; 2 ð3:26Þ
1 1 1 1 F n; n þ ; n þ ; 1 2 2 2 2 ! ! n 2n 2k X 1 k ! ¼ ð1Þ ¼ 2n k n k
2n !; 2n
n
ð3:27Þ
n
k-balanced with k ¼ 0; n ¼ 0; 1; 2; . . . ;
F
12 n; 12 n;n þ 12 ;1
¼
¼
4n Cðn2þ12Þ½Cðn2þ1Þ
3
ð2nÞ!p
P
8 <0
k ð1Þ
n k
!
2n 2k
!
n
Cðn2kþ12Þ Cðn2kþ1Þ
n odd;
: 8 Cð n
k
Þ½Cpðffiffi ð2nÞ!p p
nþ1 2
nþ1 2
Þ
;
3
n even; ð3:28Þ
1 1 1 1 1 F n þ ; n ; n þ ; 1 2 2 2 2 2 n n 1 3 n 2n 2k 4 ðn þ 1Þ C 2 þ 1 C 2 þ 2 X k n ¼ ð1Þ k ð2nÞ! p k n nþ1 n 3 n 3n2 2 C kþ1 2 nC 2 C 2 ¼ pffiffiffi 2n ; n ¼ 1; 3; 5; . . . ; 3 ð2nÞ!p p C 2kþ2 ð3:29Þ
M.E.A. El-Mikkawy, G.-S. Cheon / Appl. Math. Comput. 166 (2005) 181–195
n F ðn; n; 2n; 1Þ ¼
X k ð1Þ k
193
!2
k ! 2n k
¼
1 !; 2n
k-balanced with k ¼ 0; n ¼ 0; 1; 2; . . . ;
ð3:30Þ
n
F ðn; n þ 1; 1; 1Þ ¼
X k
n
¼ ð1Þ ;
n nþk ð1Þ k n k
k-balanced with k ¼ 0; n ¼ 0; 1; 2; . . . ;
k
n 1
ð3:31Þ
X 2 1 k F n; n þ 1; n þ 2; ¼ ðnÞ!ðn þ 1Þ! 2 ðn kÞ!ðn þ k þ 1Þ! k 1k n X 2 2n k ¼ ðn þ 1Þ ¼ ; n ¼ 0; 1; 2; . . . ; 2n þ 1 n þ k þ 1 k n
n 1
P
k ðn kÞ!ðn þ k þ 1Þ! k k n X ð1Þ k 1 ¼ ðn þ 1Þ ; n ¼ 0; 1; 2; . . . ; ¼ 2n þ 1 nþkþ1 k n
F ðn; n þ 1; n þ 2; 1Þ ¼ ðnÞ!ðn þ 1Þ!
2 6
n; n þ 1; n þ 1
3F 24
n þ 2; 1 n ð1Þ ; ¼ 2n þ 1 n
ð3:32Þ
ð3:33Þ
n n 1
3
X 7 ; 1 5 ¼ ðn þ 1Þ k
k
k nþkþ1
k-balanced with k ¼ 0; n ¼ 0; 1; 2; . . . ;
ð3:34Þ
194
M.E.A. El-Mikkawy, G.-S. Cheon / Appl. Math. Comput. 166 (2005) 181–195
21 6 6
; n; n þ 1 2
3F 24
1; 1 8 0; > < 2 ¼ Cðnþ1 2 Þ > 1 :p Cðn2þ1Þ
3 7 1 X ;17 5 ¼ pffiffiffi p k
ð1Þ
n
k
k
!
nþk
n Cðk þ 1Þ
!
C k þ 12
n ¼ 1; 3; 5; . . . ; ð3:35Þ
: n ¼ 0; 2; 4; . . . ;
n nþk X k n k F ðn; n þ 1; 2; 1Þ ¼ ¼0 ð1Þ k þ 1 k
ð3:36Þ
Saalschutzian type; n ¼ 1; 2; 3; . . . ; 21 6
3F 24
; n; n þ 1 2 3 ;1 2
n nþk
3
7 X k ;15 ¼ ð1Þ k
k
n 2k þ 1
¼
1 ; 2n þ 1
ð3:37Þ
k-balanced with k ¼ 0; n ¼ 0; 1; 2; . . . ; X k n nþk 1 1 F n; n þ 1; 1; ¼ 2 2 k n k 8 0; n odd > > < n n k-balanced with k ¼ 0; n ¼ 0; 1; 2; . . . ; ¼ ð1Þ2 n > > 2 : ; n even; 2n ð3:38Þ All results are tested using MAPLE.
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