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Fractional calculus and generalized Rodrigues formula
Applied Mathematics and Computation 147 (2004) 29–43 www.elsevier.com/locate/amc
Fractional calculus and generalized Rodrigues formula Saad Zagloul Rida a
a,*
, Ahmed M.A. El-Sayed
b
Department of Mathematics, Faculty of Science, South Valley University, Qena, Egypt b Faculty of Science, Alexandria University, Alexandria, Egypt
Abstract The topic of fractional calculus (derivatives and integrals of arbitrary orders) is enjoying growing interest not only among mathematicians, but also among physicists and engineers. The two generalizations of the Rodrigues formula of the Laguerre polynomials Lba ðxÞ ¼ ðxb =n!Þex Da ex xnþb , and Lba ðxÞ ¼ ððxb ex Þ=Cð1 þ aÞÞDa ex xaþb , are defined in [Math. Sci. Res. Hot-line 1 (10) (1997) 7; Appl. Math. Comput. 106 (1) (1999) 51] and some of their properties are proved. Here we define the new special function Lba ðc; a; xÞ based on a generalization of the Rodrigues formula, then we study some of its properties, some recurrence relations and prove that the set of functions fLba ðc; a; xÞ; a 2 Rg is continuous as a function of a 2 R. The continuation as a; c ! n and a ¼ 1 to the Rodrigues formula of the Laguerre polynomials Lbn ðxÞ are proved. Also the confluent hypergeometric representation will be given. Ó 2002 Elsevier Inc. All rights reserved.
1. Introduction Let L1 ¼ L1 ðIÞ be the class of Lebesgue integrable functions on the interval I ¼ ½a; b , where 0 6 a < b < 1, and let Cð Þ be the gamma function.
*
Corresponding author. E-mail addresses: [email protected] (S.Z. Rida), [email protected] (A.M.A. El-Sayed). 0096-3003/$ - see front matter Ó 2002 Elsevier Inc. All rights reserved. doi:10.1016/S0096-3003(02)00648-3
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S.Z. Rida, A.M.A. El-Sayed / Appl. Math. Comput. 147 (2004) 29–43
Definition 1.1. Let f ðxÞ 2 L1 , b 2 Rþ . The Rimann–Liouvill fractional (arbitrary) order integral of the function f ðxÞ of order b is defined by (see [2–8]) Z x b1 ðx sÞ b f ðsÞ ds Ia f ðxÞ ¼ CðbÞ a and when a ¼ 0, we can write I b f ðxÞ ¼ I0b f ðxÞ ¼ f ðxÞ/b ðxÞ, where /b ðxÞ ¼ xb1 =CðbÞ, for x > 0, /b ðxÞ ¼ 0, for x 6 0, and /b ! dðxÞ (the delta function), as b ! 0 (see [5]), and hence I b f ðxÞ ! f ðxÞ, as b ! 0. For the fractional order derivative we have the following definition. Definition 1.2. The fractional derivative Da of order a 2 ð0; 1Þ of the absolutely continuous function f ðxÞ is given by (see [1,6,7]) Daa f ðxÞ ¼ Ia1a Df ðxÞ;
D¼
d dx
and the fractional derivative Da of order a 2 ðn 1; nÞ of the function f ðxÞ is given by Daa f ðxÞ ¼ Iana Dn f ðxÞ;
D¼
d dx
Now we have the following two definitions. Definition 1.3. Let a; c 2 ðn 1; nÞ, n ¼ 1; 2; . . . and b; a 2 R. We define the function Yab ðc; a; xÞ by Yab ðc; a; xÞ ¼ Da xcþb eax ;
cþb>0
ð1Þ
b and the function Ya ðc; a; xÞ by b ðc; a; xÞ ¼ I a xcþb eax ; Ya
c þ b > 1
ð2Þ
Definition 1.4. Let a; c 2 ðn 1; nÞ, n ¼ 1; 2; . . . and b; a 2 R. We define the generalized Rodrigues formula by the two functions Lba ðc; a; xÞ ¼
eax xb b Y ðc; a; xÞ; Cð1 þ aÞ a
Lba ðc; a; xÞ ¼
cþb>0
eax xb b Y ðc; a; xÞ; Cð1 aÞ a
c þ b > 0
ð3Þ ð4Þ
Here we study some of properties of the functions Lba ðc; a; xÞ, and Yab ðc; a; xÞ, some recurrence relations and prove that the set of functions fLba ðc; a; xÞ; a 2 Rg is continuous as a function of a 2 R: The continuation of the function fLba ðc; a; xÞg; as a; c ! n and a ¼ 1 to the Rodrigues formula of the Laguerre
S.Z. Rida, A.M.A. El-Sayed / Appl. Math. Comput. 147 (2004) 29–43
31
polynomials Lbn ðxÞ are proved. Also we prove that the set of functions fLbm=n ðc; a; xÞ; m; n ¼ 1; 2; . . .g are orthogonal in L2 ð0; 1Þ.
2. General properties Here we prove some general properties of the two functions Lba ðc; a; xÞ, and
Yab ðc; a; xÞ
Theorem 2.1. Let a; c 2 ðn 1; nÞ, n ¼ 1; 2; . . . If b; a 2 R, then b b b ða þ 1ÞL1þa ðc; a; xÞ ¼ DLa ðc; a; xÞ þ a Lba ðc; a; xÞ x
ð5Þ
c þ b b1 La ðc; a; xÞ aLba ðc; a; xÞ x
ð6Þ
aLba1 ðc; a; xÞ ¼ ðc þ bÞLba1 ðc 1; a; xÞ aLba ðc; a; xÞ
ð7Þ
ða þ 1ÞLb1þa ðc; a; xÞ ¼
ð1 aÞLb1a ðc; a; xÞ ¼
c þ b b1 La ðc; a; xÞ aLba ðc; a; xÞ x
ð1 aÞLb1a ðc; a; xÞ ¼ ðc þ bÞLba ðc 1; a; xÞ aLba ðc; a; xÞ b b b ð1 aÞL1a ðc; a; xÞ ¼ DLa ðc; a; xÞ þ a Lba ðc; a; xÞ x
ð8Þ ð9Þ ð10Þ
1 b1 Lba ð1 þ c; a; xÞ ¼ xLbþ1 a ðc; a; xÞ ¼ La ð2 þ c; a; xÞ x
ð11Þ
aLba ð1 þ c; a; xÞ ¼ ðc þ b þ 1ÞLba1 ðc; a; xÞ aLba1 ð1 þ c; a; xÞ
ð12Þ
1 b1 Lba ð1 c; a; xÞ ¼ xLbþ1 a ðc; a; xÞ ¼ La ð2 c; a; xÞ x
ð13Þ
aLba ð1 c; a; xÞ ¼ ð1 c þ bÞLba1 ðc; a; xÞ aLba1 ð1 c; a; xÞ
ð14Þ
Proof. Differentiation Lba ðc; a; xÞ gives b DLba ðc; a; xÞ ¼ ða þ 1ÞLbaþ1 ðc; a; xÞ þ aLba ðc; a; xÞ Lba ðc; a; xÞ x Then ða þ 1ÞLb1þa ðc; a; xÞ ¼ DLba ðc; a; xÞ þ
b a Lba ðc; a; xÞ x
From the properties of the fractional calculus and the definition of Lba ðc; a; xÞ we get
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S.Z. Rida, A.M.A. El-Sayed / Appl. Math. Comput. 147 (2004) 29–43
eax xb Daþ1 xcþb eax Cð2 þ aÞ eax xb Da ðc þ bÞxcþb1 eax axcþb eax ¼ Cð2 þ aÞ 1 c þ b b1 ¼ La ðc; a; xÞ aLba ðc; a; xÞ aþ1 x
Lbaþ1 ðc; a; xÞ ¼
and eax xb Da xcþb eax Cða þ 1Þ eax xb Da1 ðc þ bÞxcþb1 eax axcþb eax ¼ Cða þ 1Þ 1 ¼ ðc þ bÞLba1 ðc 1; a; xÞ aLba1 ðc; a; xÞ a
Lba ðc; a; xÞ ¼
Also eax xb D1a xcþb eax Cð2 aÞ eax xb a I ðc þ bÞxcþb1 eax axcþb eax ¼ Cð2 aÞ 1 c þ b b1 La ðc; a; xÞ aLba ðc; a; xÞ ¼ 1a x 1 ðc þ bÞLba ðc 1; a; xÞ aLba ðc; a; xÞ ¼ 1a
Lb1a ðc; a; xÞ ¼
where 1 ðc; a; xÞ Lba ðc 1; a; xÞ ¼ Lb1 x a Since b DLba ðc; a; xÞ ¼ ð1 aÞLb1a ðc; a; xÞ þ aLba ðc; a; xÞ Lba ðc; a; xÞ x Then ð1 aÞLb1a ðc; a; xÞ ¼ DLba ðc; a; xÞ þ
b a Lba ðc; a; xÞ x
S.Z. Rida, A.M.A. El-Sayed / Appl. Math. Comput. 147 (2004) 29–43
33
From the definition of Lba ðc; a; xÞ, we get Lba ð1 þ c; a; xÞ ¼
eax xb xeax xðbþ1Þ a 1þcþb ax Da x1þcþb eax ¼ Dx e Cð1 þ aÞ Cð1 þ aÞ
eax xðb1Þ a 2þcþb1 ax Dx e ¼ xLbþ1 a ðc; a; xÞ xCð1 þ aÞ 1 ¼ Lb1 ð2 þ c; a; xÞ x a
¼
Also, we have eax xb Da x1þcþb eax Cð1 þ aÞ eax xb Da1 ðc þ b þ 1Þxcþb eax ax1þcþb eax ¼ Cð1 þ aÞ 1 ¼ ðc þ b þ 1ÞLba1 ðc; a; xÞ aLba1 ð1 þ c; a; xÞ a
Lba ð1 þ c; a; xÞ ¼
By the same way, we can easily prove the last two relation.
From the properties of the fractional calculus and the definition of Yab ðc; a; xÞ, we can easily prove the following lemma. Lemma 2.2. Let a; a1 ; c 2 ðn 1; nÞ, n ¼ 1; 2; . . . If b; a 2 R, then Da1 Yab ðc; a; xÞ ¼ Yab1 þa ðc; a; xÞ ¼ Da Yab1 ðc; a; xÞ b b b I a1 Ya ðc; a; xÞ ¼ Yða ðc; a; xÞ ¼ I a Ya ðc; a; xÞ 1 1 þaÞ b Da1 Ya ðc; a; xÞ ¼ I a Yab1 ðc; a; xÞ
3. Hypergeometric representations We study here the representation of the function Yab ðc; a; xÞ by the confluent hypergeometric functions 1 F1 ða; b; xÞ. Theorem 3.1. Let a; c 2 ðn 1; nÞ, n ¼ 1; 2; . . . If b; a 2 R, then n X n Cð1 þ c þ bÞ nk b Ya ðc; a; xÞ ¼ ð1Þ ank eax xnaþcþbk Cð1 þ n a þ c þ b kÞ k k¼0 1 F1 ðn a; 1 þ n a þ c þ b k; axÞ
ð15Þ
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S.Z. Rida, A.M.A. El-Sayed / Appl. Math. Comput. 147 (2004) 29–43
and b Ya ðc; a; xÞ ¼ eax xacþb
Cð1 c þ bÞ 1 F1 ða; 1 þ a c þ b; axÞ Cð1 þ a c þ bÞ
ð16Þ
Proof. From the properties of the fractional calculus and the definition of Yab ðc; a; xÞ we get n X n b na n cþb ax na ðDk xcþb ÞDnk eax Ya ðc; a; xÞ ¼ I D x e ¼ I k k¼0 n X Cð1 þ c þ bÞ na cþbk ax n nk I ðx ðaÞ ¼ e Þ ð17Þ k Cð1 þ c þ b kÞ k¼0 but, for n a ¼ m we have Z x 1 m1 m cþb ax ðx sÞ scþbk eas ds I x e ¼ CðmÞ 0 Z x eax m1 ¼ ðx sÞ scþbk eaðxsÞ ds CðmÞ 0 Putting x s ¼ xt, we get Z eax xmþcþbk 1 m1 cþbk axt t ð1 tÞ e dt CðmÞ 0 eax xmþcþbk Cð1 þ c þ b kÞ ¼ 1 F1 ðm; 1 þ m þ c þ b k; axÞ Cð1 þ m þ c þ b kÞ
I m xcþb eax ¼
ð18Þ Substituting from (18) into (17), we get (15). Also, from the definition of b Ya ðc; a; xÞ and the properties of fractional calculus, we have Z x 1 b Ya ðc; a; xÞ ¼ ðx sÞa1 scþb eas ds CðaÞ 0 Z x eax ¼ ðx sÞa1 scþb eaðxsÞ ds CðaÞ 0 Z eax xacþb 1 a1 ¼ t ð1 tÞcþb eaxt dt CðaÞ 0 That is b ðc; a; xÞ ¼ Ya
which proves (16).
eax xacþb Cð1 c þ bÞ 1 F1 ða; 1 þ a c þ b; axÞ Cð1 þ a c þ bÞ
S.Z. Rida, A.M.A. El-Sayed / Appl. Math. Comput. 147 (2004) 29–43
35
By the same way we can prove the following theorem. Theorem 3.2. Let a; c 2 ðn; n þ 1Þ, n ¼ 1; 2; . . . If b; a 2 R, then nþ1 X nþ1 Cð1 þ c þ bÞ nkþ1 ax 1þnaþcþbk b Ya ðc; a; xÞ ¼ e x ðaÞ Cð2 þ n a þ c þ b kÞ k k¼0 1 F1 ð1 þ n a; 2 þ n a þ c þ b k; axÞ
ð19Þ
Corollary 3.3. Let a; c 2 ðn 1; nÞ, n ¼ 1; 2; . . . If b; a 2 R, then n X n ð1Þnk ank xnaþck Cð1 þ c þ bÞ Lba ðc; a; xÞY ¼ Cð1 þ n a þ c þ b kÞ k k¼0 1 F1 ðn a; 1 þ n a þ c þ b k; axÞ
ð20Þ
and Lba ðc; a; xÞ ¼ xac
Cð1 c þ bÞ 1 F1 ða; 1 þ a c þ b; axÞ Cð1 þ a c þ bÞ
ð21Þ
Corollary 3.4. Let a; c 2 ðn; n þ 1Þ, n ¼ 1; 2; . . . If b; a 2 R, then nþ1 X n þ 1 ðaÞnkþ1 x1þnaþck Cð1 þ c þ bÞ Lba ðc; a; xÞ ¼ Cð2 þ n a þ c þ b kÞ k k¼0 1 F1 ð1 þ n a; 2 þ n a þ c þ b k; axÞ
ð22Þ
4. Continuation properties Here we study the continuity properties with respect to a of the set of functions fYab ðc; a; xÞ; a 2 Rg and fLba ðc; a; xÞ; a 2 Rg. Theorem 4.1. If a; b 2 R, then lim Yab ðc; a; xÞ ¼ Ynb ðc; a; xÞ ¼ limþ Yab ðc; a; xÞ
a!n
a!n
Proof. Let a; c 2 ðn 1; nÞ. Since 1 F1 ðn
a; 1 þ n a þ c þ b k; axÞ
¼ eax F1 ð1 þ c þ b k; 1 þ n a þ c þ b k; axÞ and lim 1 F1 ð1 þ c þ b k; 1 þ n a þ c þ b k; axÞ ¼ eax
a!n
ð23Þ
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S.Z. Rida, A.M.A. El-Sayed / Appl. Math. Comput. 147 (2004) 29–43
then substituting in (15) and taking the limit as a ! n , we obtain n X Cð1 þ c þ bÞ cþbk ax n nk x lim Yab ðc; a; xÞ ¼ ðaÞ e ¼ Ynb ðc; a; xÞ k a!n Cð1 þ c þ b kÞ k¼0 ð24Þ Now, let c; a 2 ðn; n þ 1Þ, then from (19), we have limþ Yab ðc; a; xÞ ¼
a!n
nþ1 X nþ1 ðaÞ1þnk eax x1þcþbk k k¼0
Cð1 þ c þ bÞ 1 F1 ð1; 2 þ c þ b k; axÞ Cð2 þ c þ b kÞ
Substituting the value of 1 F1 ð1; 2 þ c þ b k; axÞ we get limþ Yab ðc; a; xÞ ¼
a!n
nþ1 X nþ1 1þnk ax 1þcþbk ðaÞ e x k k¼0
r 1 Cð1 þ c þ bÞ X ðaxÞ Cð2 þ c þ b kÞ r¼0 Cð2 þ c þ b k þ rÞ nþ1 X nþ1 1þnk ax b ¼ e x Cð1 þ c þ bÞ ðaÞ k k¼0
r 1 X x1þck ðaxÞ Cð2 þ c þ b kÞ r¼0 Cð2 þ c þ b k þ rÞ
ð25Þ
Interchanging the order of the two summations and using the properties of the binomial coefficients, we get ( ! nþ1 nþ1 b ax lim Ya ðc; a; xÞ ¼ e Cð1 þ c þ bÞ ð aÞ x1þcþb a!nþ 0 ! r 1 X nþ1 ðaxÞ n þ ð aÞ xcþb Cð2 þ c þ b þ rÞ 1 r¼0 " # ! r 1 X nþ1 c ðaxÞ þ þ
þ Cð1 þ c þ bÞ r¼1 Cð1 þ c þ b þ rÞ nþ1 " 1 ax 0 þ
þ ð aÞ xcþbn Cð1 þ c n þ bÞ Cð2 þ c n þ bÞ #) n r 1 X ðaxÞ ðaxÞ þ þ Cð1 þ c þ bÞ r¼nþ1 Cð2 þ c þ b ðn þ 1Þ þ rÞ
S.Z. Rida, A.M.A. El-Sayed / Appl. Math. Comput. 147 (2004) 29–43
37
Hence 1 X a1þnþr x1þcþbþr Cð2 þ c þ b þ rÞ r¼0 r n X ðaxÞ n n þ eax xcþb ðaÞ Cð1 þ c þ bÞ r Cð1 þ c þ b rÞ r¼0 nþ1
limþ Yab ðc; a; xÞ ¼ eax Cð1 þ c þ bÞð1 1Þ
a!n
which proves that limþ Yab ðc; a; xÞ ¼
a!n
n X
ðaÞ
nr
r¼0
Cð1 þ c þ bÞ cþbr ax x e ¼ Ynb ðc; a; xÞ Cð1 þ c þ b rÞ ð26Þ
Combining (24) and (26), we deduce that lim Yab ðc; a; xÞ ¼ limþ Yab ðc; a; xÞ ¼ Ynb ðc; a; xÞ
a!n
ð27Þ
a!n
Corollary 4.2. If a; b 2 R, then lim Lba ðc; a; xÞ ¼ Lbn ðc; a; xÞ ¼ limþ Lba ðc; a; xÞ
a!n
ð28Þ
a!n
Also from the properties of the gamma and hypergeometric functions, we have the following theorem. Theorem 4.3. Let a; c 2 ðn 1; nÞ, n ¼ 1; 2; . . . and b; a 2 R, then b b b lim Ya ðc; a; xÞ ¼ Yn ðc; a; xÞ ¼ limþ Ya ðc; a; xÞ
a!n
ð29Þ
a!n
where b Ya ðc; a; xÞ ¼
eax xncþb Cð1 c þ bÞ 1 F1 ðn; 1 þ n c þ b; axÞ Cð1 þ n c þ bÞ
Proof. Taking the limits as a ! n ; nþ respectively we get the result.
Corollary 4.4. If a; b 2 R, then lim Lba ðc; a; xÞ ¼ Lbn ðc; a; xÞ ¼ limþ Lba ðc; a; xÞ
a!n
a!n
ð30Þ
Lemma 4.5. Let a; c 2 ð0; 1Þ and b; a 2 R, then lim Lba ðc; a; xÞ ¼ xc ¼ lim Lba ðc; a; xÞ
a!0þ
a!0
ð31Þ
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S.Z. Rida, A.M.A. El-Sayed / Appl. Math. Comput. 147 (2004) 29–43
and lim Lba ðc; a; xÞ ¼ xc ¼ lim Lba ðc; a; xÞ
a!0þ
ð32Þ
a!0
Corollary 4.6. If a ¼ c, a ¼ 1, then lim Lba ða; 1; xÞ ¼ lim Lba ða; 1; xÞ ¼ 1 ¼ limþ Lba ða; 1; xÞ ¼ lim Lba ða; 1; xÞ
a!0þ
a!0
a!0
a!0
ð33Þ Let us first recall some known facts. The abstract Bell polynomials Bn ¼ Bn ðx1 ; x2 ; . . . ; xn Þ (n ¼ 1; 2; . . .) in variables x1 ; x2 ; . . . ; xn are given by the recursion Bnþ1 ¼
n X n k¼0
k
Bk xnþ1k
for n ¼ 0; 1; . . . together with B0 ¼ 1, while the nth Bell differential polynomials, is the result of n times applications of D þ u, D ¼ d=dx to the constant function 1: Bn ðu; u0 ; . . . ; uðn1Þ Þ ¼ ðD þ uÞn 1 The two definitions are equivalent to each other, cf. [9–11], and are valid in the matrix case too, where x1 ; x2 ; . . . ; xn or u ¼ uðxÞ respectively have values in an algebra of matrices. Now, we define the function Yab ðc; a; xÞ by the use of the Bell polynomials Bn ðxÞ, n ¼ 0; 1; 2; . . . (see [9–11]). Definition 4.7. Let a; c 2 ðn 1; nÞ, n ¼ 1; 2; . . . and b; a 2 R; a > 0. The function Yab ðc; a; xÞ of order a is defined by Yab ðc; a; xÞ
¼I
na
n X n k¼0
k
Cð1 þ c þ bÞ eax Bk ða; 0; . . . ; 0Þxcnþbþk Cð1 þ c n þ b þ kÞ ð34Þ
By this new definition, we can prove the following lemma Lemma 4.8. Let a; c 2 ðn 1; nÞ, n ¼ 1; 2; . . . and b; a 2 R; a > 0. Then lim Yab ðc; a; xÞ ¼ Ynb ðc; a; xÞ a!n
ð35Þ
S.Z. Rida, A.M.A. El-Sayed / Appl. Math. Comput. 147 (2004) 29–43
39
Proof. ( lim Yab ðc; a; xÞ a!n
¼ lim I
n X n
na
a!n
k
k¼0
Cð1 þ c þ bÞ Cð1 þ c n þ b þ kÞ )
eax Bk ð a; 0; . . . ; 0Þxcnþbþk ( ¼ lim /na ðxÞ a!n
n X n k¼0
k
Cð1 þ c þ bÞ Cð1 þ c n þ b þ kÞ )
eax Bk ð a; 0; . . . ; 0Þxcnþbþk ¼ /0 ðxÞ
n X n
k
k¼0
Cð1 þ c þ bÞ Cð1 þ c n þ b þ kÞ
ax
¼
e Bk ða; 0; . . . ; 0Þxcnþbþk n X n Cð1 þ c þ bÞ k¼0
k
Cð1 þ c n þ b þ kÞ
eax Bk ða; 0; . . . ; 0Þxcnþbþk ¼ Ynb ðc; a; xÞ
5. Recurrence relations and the differential equation We now give some recurrence relations for the functions Yab ðc; a; xÞ, Lba ðc; a; xÞ and we prove that these functions are particular solutions of second order differential equations, respectively xD2 Yab ðc; a; xÞ þ ð1 þ a c b þ axÞDYab ðc; a; xÞ þ ð1 þ aÞaYab ðc; a; xÞ ¼ 0 xD2 Lba ðc; a; xÞ þ ð1 þ a þ b c axÞDLba ðc; a; xÞ ða cÞb b La ðc; a; xÞ ¼ 0 þ ac þ x
Theorem 5.1. Let a; c 2 ðn 1; nÞ, n ¼ 1; 2; . . . and b; a 2 R; a > 0. Then b1 b Yab ðc; a; xÞ ¼ ðc þ bÞYa1 ðc; a; xÞ aYa1 ðc; a; xÞ b b ¼ ðc þ bÞYa1 ðc 1; a; xÞ aYa1 ðc; a; xÞ
ð36Þ
40
S.Z. Rida, A.M.A. El-Sayed / Appl. Math. Comput. 147 (2004) 29–43 b DYab ðc; a; xÞ ¼ Yaþ1 ðc; a; xÞ ¼ ðc þ bÞYab1 ðc; a; xÞ aYab ðc; a; xÞ b ¼ ðc þ bÞDYa1 ðc 1; a; xÞ aYab ðc; a; xÞ
ð37Þ
b ðc; a; xÞ Dn Yab ðc; a; xÞ ¼ Yaþn n X n Cð1 þ c þ bÞ nk Yabk ðc; a; xÞ ¼ ðaÞ k Cð1 þ c þ b kÞ k¼0
ð38Þ
b b xDYa1 ðc 1; a; xÞ Yab ðc; a; xÞ ¼ aYa1 ðc 1; a; xÞ
ð39Þ
Proof. From the definition of Yab ðc; a; xÞ, we have Yab ðc; a; xÞ ¼ Da1 ðc þ bÞxcþb1 eax axcþb eax ¼ ðc þ bÞDa1 xðc1Þþb eax aDa1 xcþb eax b1 b ¼ ðc þ bÞYa1 ðc; a; xÞ aYa1 ðc; a; xÞ
Also, DYab ðc; a; xÞ ¼ Da ðc þ bÞxcþb1 eax axcþb eax ¼ ðc þ bÞYab1 ðc; a; xÞ aYab ðc; a; xÞ b b ¼ ðc þ bÞDYa1 ðc 1; a; xÞ aYa1 ðc; a; xÞ
where b1 DYa1 ðc; a; xÞ ¼ Yab1 ðc; a; xÞ
and also n
D
Yab ðc; a; xÞ
n X n ðDk xcþb ÞðDnk eax Þ ¼D k k¼0 n X Cð1 þ c þ bÞ n nk Da xcþbk eax ðaÞ ¼ k Cð1 þ c þ b kÞ k¼0 n X Cð1 þ c þ bÞ n Y bk ðc; a; xÞ ¼ ðaÞnk k Cð1 þ c þ b kÞ a k¼0 a
From the convergence of the power series expansion of eax xcþb and the properties of the fractional derivative we obtain Yab ðc; a; xÞ ¼
m 1 X ðaÞ Cð1 þ m þ c þ bÞ mþcaþb x Cð1 þ m þ c a þ bÞ m! m¼0
from which we can prove (by direct substitution) the last result.
ð40Þ
S.Z. Rida, A.M.A. El-Sayed / Appl. Math. Comput. 147 (2004) 29–43
41
Theorem 5.2. Let a; c 2 ðn 1; nÞ, n ¼ 1; 2; . . . and b; a 2 R; a > 0. Then b xDLba ðc; a; xÞ ¼ ðc þ bÞLb1 a ðc; a; xÞ þ ð2ax bÞLa ðc; a; xÞ
þ abLba ðc; a; xÞ ¼ ðc þ bÞ axÞLba1 ðc 1; a; xÞ þ xDLba1 ðc
ð41Þ
axDLba ðc; a; xÞ
1; a; xÞ n Cða þ 1Þ X n k bk k! x Lk ðxÞ½Dk Lba ðc; a; xÞ Lbaþn ðc; a; xÞ ¼ k Cða þ n þ 1Þ k¼0 ½ðb
ð42Þ ð43Þ
where Lbk k ðxÞ is the wellknown Laguerre polynomials. xDLba1 ðc 1; a; xÞ aLba ðc; a; xÞ ¼ ðax b aÞLba1 ðc 1; a; xÞ
ð44Þ
Proof. From the definition of Lba ðc; a; xÞ and Eqs. (36)–(39) respectively, we have the result. Theorem 5.3. The function Yab ðc; a; xÞ is particular solution of the differential equation xD2 Yab ðc; a; xÞ þ ð1 þ a c b þ axÞDYab ðc; a; xÞ þ ð1 þ aÞaYab ðc; a; xÞ ¼ 0
ð45Þ
Proof. Multiplying (37) by x and from (39) into it we have xDYab ðc; a; xÞ ¼ axYab ðc; a; xÞ þ ðc þ bÞYab ðc; a; xÞ b ðc 1; a; xÞ aðc þ bÞYa1
ð46Þ
Differentiating this equation and from (37) into it, we obtain the result.
Theorem 5.4. The function Lba ðc; a; xÞ is particular solution of the differential equation xD2 Lba ðc; a; xÞ þ ð1 þ a þ b c axÞDLba ðc; a; xÞ ða cÞb b La ðc; a; xÞ ¼ 0 þ ac þ x Proof. From (3) into (45), we obtain the result.
ð47Þ
6. Orthogonality property Theorem 6.1. For any real numbers a1 6¼ a2 , c1 6¼ c2 , we have Z 1 eax xb Lba1 ðc1 ; a; xÞLba2 ðc2 ; a; xÞ dx ¼ 0 0
ð48Þ
42
S.Z. Rida, A.M.A. El-Sayed / Appl. Math. Comput. 147 (2004) 29–43
for a1 ¼ c1 , a2 ¼ c2 . Proof. Let uba ðc; a; xÞ ¼ eax=2 xb=2 Lba ðc; a; xÞ, a 2 Rþ , then by direct calculation we can prove that xD2 uba ðc; a; xÞ þ ð1 þ a cÞDuba ðc; a; xÞ a b b2 a2 þ ð1 þ a þ c þ bÞ þ ða cÞ x uba ðc; a; xÞ ¼ 0 2 2x 4x 4
ð49Þ
Then for any positive real numbers a1 ; c1 and a2 ; c2 , we have xD2 uba1 ðc1 ; a; xÞ þ ð1 þ a1 c1 ÞDuba1 ðc1 ; a; xÞ a b b2 a2 þ ð1 þ a1 þ c1 þ bÞ þ ða1 c1 Þ x uba1 ðc1 ; a; xÞ ¼ 0 ð50Þ 2 2x 4x 4 xD2 uba2 ðc2 ; a; xÞ þ ð1 þ a2 c2 ÞDuba2 ðc2 ; a; xÞ a b b2 a2 þ ð1 þ a2 þ c2 þ bÞ þ ða2 c2 Þ x uba2 ðc2 ; a; xÞ ¼ 0 ð51Þ 2 2x 4x 4 By multiplying the first equation by uba2 ðc2 ; a; xÞ and the second equation by uba1 ðc1 ; a; xÞ, subtracting the resulting equations and integrating from 0 to 1 we obtain Z 1 Z 1 uba1 ðc1 ; a; xÞuba2 ðc2 ; a; xÞ dx ¼ ðc1 a1 Þ uba2 ðc2 ; a; xÞDuba1 ðc1 ; a; xÞ dx 0 0 Z 1 þ ða2 c2 Þ uba1 ðc1 ; a; xÞDuba2 ðc2 ; a; xÞ dx 0
ð52Þ Then, from which, for a1 ¼ c1 and a2 ¼ c2 , we have the result.
References [1] M. Caputo, Linear model of dissipation whose Q is almost frequency independent II, Geophys. J.R. Astron. Soc. 13 (1967) 529–539. [2] A.M.A. El-Sayed, Fractional calculus and Laguerre polynomials of fractional orders, Math. Sci. Res. Hot-line 1 (10) (1997) 7–14. [3] A.M.A. El-Sayed, S.Z. Rida, Bell polynomials of arbitrary (fractional) orders, Appl. Math. Comput. 106 (1) (1999) 51–62. [4] A.M.A. El-Sayed, Laguerre polynomials of arbitrary (fractional) orders, Appl. Math. Comput. 109 (1) (2000) 1–9. [5] I.M. Gelfand, G.E. Shilov, in: Generalized Functions, vol. 1, 1958, Moscow. [6] K.S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, 1993.
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[7] I. Podlubny, A.M.A. El-Sayed, On two definitions of fractional calculus, Slovak Academy of Sciences Institute of Experimental Physics, UEF-03-96, ISBN 80-7099-252-2, 1996. [8] I. Podlubny, Fractional Differential Equation, Academic Press, San Diego, 1999. [9] R. Schimming, S.Z. Rida, Non-commutative Bell polynomials, Int. J. Algebra Comput. 6 (1996) 635–644. [10] R. Schimming, S.Z. Rida, The Bell differential polynomials, in: G.E. Bergum, et al. (Eds.), Applications of Fibonacci Numbers, vol. 7, Kluwer, Dordrecht, 1998, pp. 353–367. [11] S.Z. Rida, Powers of differential operators and explicit differential polynomials, Ph.D. Thesis, Free University Berlin, ISBN 3-8265-1670-2, 1996.
Fractional calculus and generalized Rodrigues formula Saad Zagloul Rida a
a,*
, Ahmed M.A. El-Sayed
b
Department of Mathematics, Faculty of Science, South Valley University, Qena, Egypt b Faculty of Science, Alexandria University, Alexandria, Egypt
Abstract The topic of fractional calculus (derivatives and integrals of arbitrary orders) is enjoying growing interest not only among mathematicians, but also among physicists and engineers. The two generalizations of the Rodrigues formula of the Laguerre polynomials Lba ðxÞ ¼ ðxb =n!Þex Da ex xnþb , and Lba ðxÞ ¼ ððxb ex Þ=Cð1 þ aÞÞDa ex xaþb , are defined in [Math. Sci. Res. Hot-line 1 (10) (1997) 7; Appl. Math. Comput. 106 (1) (1999) 51] and some of their properties are proved. Here we define the new special function Lba ðc; a; xÞ based on a generalization of the Rodrigues formula, then we study some of its properties, some recurrence relations and prove that the set of functions fLba ðc; a; xÞ; a 2 Rg is continuous as a function of a 2 R. The continuation as a; c ! n and a ¼ 1 to the Rodrigues formula of the Laguerre polynomials Lbn ðxÞ are proved. Also the confluent hypergeometric representation will be given. Ó 2002 Elsevier Inc. All rights reserved.
1. Introduction Let L1 ¼ L1 ðIÞ be the class of Lebesgue integrable functions on the interval I ¼ ½a; b , where 0 6 a < b < 1, and let Cð Þ be the gamma function.
*
Corresponding author. E-mail addresses: [email protected] (S.Z. Rida), [email protected] (A.M.A. El-Sayed). 0096-3003/$ - see front matter Ó 2002 Elsevier Inc. All rights reserved. doi:10.1016/S0096-3003(02)00648-3
30
S.Z. Rida, A.M.A. El-Sayed / Appl. Math. Comput. 147 (2004) 29–43
Definition 1.1. Let f ðxÞ 2 L1 , b 2 Rþ . The Rimann–Liouvill fractional (arbitrary) order integral of the function f ðxÞ of order b is defined by (see [2–8]) Z x b1 ðx sÞ b f ðsÞ ds Ia f ðxÞ ¼ CðbÞ a and when a ¼ 0, we can write I b f ðxÞ ¼ I0b f ðxÞ ¼ f ðxÞ/b ðxÞ, where /b ðxÞ ¼ xb1 =CðbÞ, for x > 0, /b ðxÞ ¼ 0, for x 6 0, and /b ! dðxÞ (the delta function), as b ! 0 (see [5]), and hence I b f ðxÞ ! f ðxÞ, as b ! 0. For the fractional order derivative we have the following definition. Definition 1.2. The fractional derivative Da of order a 2 ð0; 1Þ of the absolutely continuous function f ðxÞ is given by (see [1,6,7]) Daa f ðxÞ ¼ Ia1a Df ðxÞ;
D¼
d dx
and the fractional derivative Da of order a 2 ðn 1; nÞ of the function f ðxÞ is given by Daa f ðxÞ ¼ Iana Dn f ðxÞ;
D¼
d dx
Now we have the following two definitions. Definition 1.3. Let a; c 2 ðn 1; nÞ, n ¼ 1; 2; . . . and b; a 2 R. We define the function Yab ðc; a; xÞ by Yab ðc; a; xÞ ¼ Da xcþb eax ;
cþb>0
ð1Þ
b and the function Ya ðc; a; xÞ by b ðc; a; xÞ ¼ I a xcþb eax ; Ya
c þ b > 1
ð2Þ
Definition 1.4. Let a; c 2 ðn 1; nÞ, n ¼ 1; 2; . . . and b; a 2 R. We define the generalized Rodrigues formula by the two functions Lba ðc; a; xÞ ¼
eax xb b Y ðc; a; xÞ; Cð1 þ aÞ a
Lba ðc; a; xÞ ¼
cþb>0
eax xb b Y ðc; a; xÞ; Cð1 aÞ a
c þ b > 0
ð3Þ ð4Þ
Here we study some of properties of the functions Lba ðc; a; xÞ, and Yab ðc; a; xÞ, some recurrence relations and prove that the set of functions fLba ðc; a; xÞ; a 2 Rg is continuous as a function of a 2 R: The continuation of the function fLba ðc; a; xÞg; as a; c ! n and a ¼ 1 to the Rodrigues formula of the Laguerre
S.Z. Rida, A.M.A. El-Sayed / Appl. Math. Comput. 147 (2004) 29–43
31
polynomials Lbn ðxÞ are proved. Also we prove that the set of functions fLbm=n ðc; a; xÞ; m; n ¼ 1; 2; . . .g are orthogonal in L2 ð0; 1Þ.
2. General properties Here we prove some general properties of the two functions Lba ðc; a; xÞ, and
Yab ðc; a; xÞ
Theorem 2.1. Let a; c 2 ðn 1; nÞ, n ¼ 1; 2; . . . If b; a 2 R, then b b b ða þ 1ÞL1þa ðc; a; xÞ ¼ DLa ðc; a; xÞ þ a Lba ðc; a; xÞ x
ð5Þ
c þ b b1 La ðc; a; xÞ aLba ðc; a; xÞ x
ð6Þ
aLba1 ðc; a; xÞ ¼ ðc þ bÞLba1 ðc 1; a; xÞ aLba ðc; a; xÞ
ð7Þ
ða þ 1ÞLb1þa ðc; a; xÞ ¼
ð1 aÞLb1a ðc; a; xÞ ¼
c þ b b1 La ðc; a; xÞ aLba ðc; a; xÞ x
ð1 aÞLb1a ðc; a; xÞ ¼ ðc þ bÞLba ðc 1; a; xÞ aLba ðc; a; xÞ b b b ð1 aÞL1a ðc; a; xÞ ¼ DLa ðc; a; xÞ þ a Lba ðc; a; xÞ x
ð8Þ ð9Þ ð10Þ
1 b1 Lba ð1 þ c; a; xÞ ¼ xLbþ1 a ðc; a; xÞ ¼ La ð2 þ c; a; xÞ x
ð11Þ
aLba ð1 þ c; a; xÞ ¼ ðc þ b þ 1ÞLba1 ðc; a; xÞ aLba1 ð1 þ c; a; xÞ
ð12Þ
1 b1 Lba ð1 c; a; xÞ ¼ xLbþ1 a ðc; a; xÞ ¼ La ð2 c; a; xÞ x
ð13Þ
aLba ð1 c; a; xÞ ¼ ð1 c þ bÞLba1 ðc; a; xÞ aLba1 ð1 c; a; xÞ
ð14Þ
Proof. Differentiation Lba ðc; a; xÞ gives b DLba ðc; a; xÞ ¼ ða þ 1ÞLbaþ1 ðc; a; xÞ þ aLba ðc; a; xÞ Lba ðc; a; xÞ x Then ða þ 1ÞLb1þa ðc; a; xÞ ¼ DLba ðc; a; xÞ þ
b a Lba ðc; a; xÞ x
From the properties of the fractional calculus and the definition of Lba ðc; a; xÞ we get
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S.Z. Rida, A.M.A. El-Sayed / Appl. Math. Comput. 147 (2004) 29–43
eax xb Daþ1 xcþb eax Cð2 þ aÞ eax xb Da ðc þ bÞxcþb1 eax axcþb eax ¼ Cð2 þ aÞ 1 c þ b b1 ¼ La ðc; a; xÞ aLba ðc; a; xÞ aþ1 x
Lbaþ1 ðc; a; xÞ ¼
and eax xb Da xcþb eax Cða þ 1Þ eax xb Da1 ðc þ bÞxcþb1 eax axcþb eax ¼ Cða þ 1Þ 1 ¼ ðc þ bÞLba1 ðc 1; a; xÞ aLba1 ðc; a; xÞ a
Lba ðc; a; xÞ ¼
Also eax xb D1a xcþb eax Cð2 aÞ eax xb a I ðc þ bÞxcþb1 eax axcþb eax ¼ Cð2 aÞ 1 c þ b b1 La ðc; a; xÞ aLba ðc; a; xÞ ¼ 1a x 1 ðc þ bÞLba ðc 1; a; xÞ aLba ðc; a; xÞ ¼ 1a
Lb1a ðc; a; xÞ ¼
where 1 ðc; a; xÞ Lba ðc 1; a; xÞ ¼ Lb1 x a Since b DLba ðc; a; xÞ ¼ ð1 aÞLb1a ðc; a; xÞ þ aLba ðc; a; xÞ Lba ðc; a; xÞ x Then ð1 aÞLb1a ðc; a; xÞ ¼ DLba ðc; a; xÞ þ
b a Lba ðc; a; xÞ x
S.Z. Rida, A.M.A. El-Sayed / Appl. Math. Comput. 147 (2004) 29–43
33
From the definition of Lba ðc; a; xÞ, we get Lba ð1 þ c; a; xÞ ¼
eax xb xeax xðbþ1Þ a 1þcþb ax Da x1þcþb eax ¼ Dx e Cð1 þ aÞ Cð1 þ aÞ
eax xðb1Þ a 2þcþb1 ax Dx e ¼ xLbþ1 a ðc; a; xÞ xCð1 þ aÞ 1 ¼ Lb1 ð2 þ c; a; xÞ x a
¼
Also, we have eax xb Da x1þcþb eax Cð1 þ aÞ eax xb Da1 ðc þ b þ 1Þxcþb eax ax1þcþb eax ¼ Cð1 þ aÞ 1 ¼ ðc þ b þ 1ÞLba1 ðc; a; xÞ aLba1 ð1 þ c; a; xÞ a
Lba ð1 þ c; a; xÞ ¼
By the same way, we can easily prove the last two relation.
From the properties of the fractional calculus and the definition of Yab ðc; a; xÞ, we can easily prove the following lemma. Lemma 2.2. Let a; a1 ; c 2 ðn 1; nÞ, n ¼ 1; 2; . . . If b; a 2 R, then Da1 Yab ðc; a; xÞ ¼ Yab1 þa ðc; a; xÞ ¼ Da Yab1 ðc; a; xÞ b b b I a1 Ya ðc; a; xÞ ¼ Yða ðc; a; xÞ ¼ I a Ya ðc; a; xÞ 1 1 þaÞ b Da1 Ya ðc; a; xÞ ¼ I a Yab1 ðc; a; xÞ
3. Hypergeometric representations We study here the representation of the function Yab ðc; a; xÞ by the confluent hypergeometric functions 1 F1 ða; b; xÞ. Theorem 3.1. Let a; c 2 ðn 1; nÞ, n ¼ 1; 2; . . . If b; a 2 R, then n X n Cð1 þ c þ bÞ nk b Ya ðc; a; xÞ ¼ ð1Þ ank eax xnaþcþbk Cð1 þ n a þ c þ b kÞ k k¼0 1 F1 ðn a; 1 þ n a þ c þ b k; axÞ
ð15Þ
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S.Z. Rida, A.M.A. El-Sayed / Appl. Math. Comput. 147 (2004) 29–43
and b Ya ðc; a; xÞ ¼ eax xacþb
Cð1 c þ bÞ 1 F1 ða; 1 þ a c þ b; axÞ Cð1 þ a c þ bÞ
ð16Þ
Proof. From the properties of the fractional calculus and the definition of Yab ðc; a; xÞ we get n X n b na n cþb ax na ðDk xcþb ÞDnk eax Ya ðc; a; xÞ ¼ I D x e ¼ I k k¼0 n X Cð1 þ c þ bÞ na cþbk ax n nk I ðx ðaÞ ¼ e Þ ð17Þ k Cð1 þ c þ b kÞ k¼0 but, for n a ¼ m we have Z x 1 m1 m cþb ax ðx sÞ scþbk eas ds I x e ¼ CðmÞ 0 Z x eax m1 ¼ ðx sÞ scþbk eaðxsÞ ds CðmÞ 0 Putting x s ¼ xt, we get Z eax xmþcþbk 1 m1 cþbk axt t ð1 tÞ e dt CðmÞ 0 eax xmþcþbk Cð1 þ c þ b kÞ ¼ 1 F1 ðm; 1 þ m þ c þ b k; axÞ Cð1 þ m þ c þ b kÞ
I m xcþb eax ¼
ð18Þ Substituting from (18) into (17), we get (15). Also, from the definition of b Ya ðc; a; xÞ and the properties of fractional calculus, we have Z x 1 b Ya ðc; a; xÞ ¼ ðx sÞa1 scþb eas ds CðaÞ 0 Z x eax ¼ ðx sÞa1 scþb eaðxsÞ ds CðaÞ 0 Z eax xacþb 1 a1 ¼ t ð1 tÞcþb eaxt dt CðaÞ 0 That is b ðc; a; xÞ ¼ Ya
which proves (16).
eax xacþb Cð1 c þ bÞ 1 F1 ða; 1 þ a c þ b; axÞ Cð1 þ a c þ bÞ
S.Z. Rida, A.M.A. El-Sayed / Appl. Math. Comput. 147 (2004) 29–43
35
By the same way we can prove the following theorem. Theorem 3.2. Let a; c 2 ðn; n þ 1Þ, n ¼ 1; 2; . . . If b; a 2 R, then nþ1 X nþ1 Cð1 þ c þ bÞ nkþ1 ax 1þnaþcþbk b Ya ðc; a; xÞ ¼ e x ðaÞ Cð2 þ n a þ c þ b kÞ k k¼0 1 F1 ð1 þ n a; 2 þ n a þ c þ b k; axÞ
ð19Þ
Corollary 3.3. Let a; c 2 ðn 1; nÞ, n ¼ 1; 2; . . . If b; a 2 R, then n X n ð1Þnk ank xnaþck Cð1 þ c þ bÞ Lba ðc; a; xÞY ¼ Cð1 þ n a þ c þ b kÞ k k¼0 1 F1 ðn a; 1 þ n a þ c þ b k; axÞ
ð20Þ
and Lba ðc; a; xÞ ¼ xac
Cð1 c þ bÞ 1 F1 ða; 1 þ a c þ b; axÞ Cð1 þ a c þ bÞ
ð21Þ
Corollary 3.4. Let a; c 2 ðn; n þ 1Þ, n ¼ 1; 2; . . . If b; a 2 R, then nþ1 X n þ 1 ðaÞnkþ1 x1þnaþck Cð1 þ c þ bÞ Lba ðc; a; xÞ ¼ Cð2 þ n a þ c þ b kÞ k k¼0 1 F1 ð1 þ n a; 2 þ n a þ c þ b k; axÞ
ð22Þ
4. Continuation properties Here we study the continuity properties with respect to a of the set of functions fYab ðc; a; xÞ; a 2 Rg and fLba ðc; a; xÞ; a 2 Rg. Theorem 4.1. If a; b 2 R, then lim Yab ðc; a; xÞ ¼ Ynb ðc; a; xÞ ¼ limþ Yab ðc; a; xÞ
a!n
a!n
Proof. Let a; c 2 ðn 1; nÞ. Since 1 F1 ðn
a; 1 þ n a þ c þ b k; axÞ
¼ eax F1 ð1 þ c þ b k; 1 þ n a þ c þ b k; axÞ and lim 1 F1 ð1 þ c þ b k; 1 þ n a þ c þ b k; axÞ ¼ eax
a!n
ð23Þ
36
S.Z. Rida, A.M.A. El-Sayed / Appl. Math. Comput. 147 (2004) 29–43
then substituting in (15) and taking the limit as a ! n , we obtain n X Cð1 þ c þ bÞ cþbk ax n nk x lim Yab ðc; a; xÞ ¼ ðaÞ e ¼ Ynb ðc; a; xÞ k a!n Cð1 þ c þ b kÞ k¼0 ð24Þ Now, let c; a 2 ðn; n þ 1Þ, then from (19), we have limþ Yab ðc; a; xÞ ¼
a!n
nþ1 X nþ1 ðaÞ1þnk eax x1þcþbk k k¼0
Cð1 þ c þ bÞ 1 F1 ð1; 2 þ c þ b k; axÞ Cð2 þ c þ b kÞ
Substituting the value of 1 F1 ð1; 2 þ c þ b k; axÞ we get limþ Yab ðc; a; xÞ ¼
a!n
nþ1 X nþ1 1þnk ax 1þcþbk ðaÞ e x k k¼0
r 1 Cð1 þ c þ bÞ X ðaxÞ Cð2 þ c þ b kÞ r¼0 Cð2 þ c þ b k þ rÞ nþ1 X nþ1 1þnk ax b ¼ e x Cð1 þ c þ bÞ ðaÞ k k¼0
r 1 X x1þck ðaxÞ Cð2 þ c þ b kÞ r¼0 Cð2 þ c þ b k þ rÞ
ð25Þ
Interchanging the order of the two summations and using the properties of the binomial coefficients, we get ( ! nþ1 nþ1 b ax lim Ya ðc; a; xÞ ¼ e Cð1 þ c þ bÞ ð aÞ x1þcþb a!nþ 0 ! r 1 X nþ1 ðaxÞ n þ ð aÞ xcþb Cð2 þ c þ b þ rÞ 1 r¼0 " # ! r 1 X nþ1 c ðaxÞ þ þ
þ Cð1 þ c þ bÞ r¼1 Cð1 þ c þ b þ rÞ nþ1 " 1 ax 0 þ
þ ð aÞ xcþbn Cð1 þ c n þ bÞ Cð2 þ c n þ bÞ #) n r 1 X ðaxÞ ðaxÞ þ þ Cð1 þ c þ bÞ r¼nþ1 Cð2 þ c þ b ðn þ 1Þ þ rÞ
S.Z. Rida, A.M.A. El-Sayed / Appl. Math. Comput. 147 (2004) 29–43
37
Hence 1 X a1þnþr x1þcþbþr Cð2 þ c þ b þ rÞ r¼0 r n X ðaxÞ n n þ eax xcþb ðaÞ Cð1 þ c þ bÞ r Cð1 þ c þ b rÞ r¼0 nþ1
limþ Yab ðc; a; xÞ ¼ eax Cð1 þ c þ bÞð1 1Þ
a!n
which proves that limþ Yab ðc; a; xÞ ¼
a!n
n X
ðaÞ
nr
r¼0
Cð1 þ c þ bÞ cþbr ax x e ¼ Ynb ðc; a; xÞ Cð1 þ c þ b rÞ ð26Þ
Combining (24) and (26), we deduce that lim Yab ðc; a; xÞ ¼ limþ Yab ðc; a; xÞ ¼ Ynb ðc; a; xÞ
a!n
ð27Þ
a!n
Corollary 4.2. If a; b 2 R, then lim Lba ðc; a; xÞ ¼ Lbn ðc; a; xÞ ¼ limþ Lba ðc; a; xÞ
a!n
ð28Þ
a!n
Also from the properties of the gamma and hypergeometric functions, we have the following theorem. Theorem 4.3. Let a; c 2 ðn 1; nÞ, n ¼ 1; 2; . . . and b; a 2 R, then b b b lim Ya ðc; a; xÞ ¼ Yn ðc; a; xÞ ¼ limþ Ya ðc; a; xÞ
a!n
ð29Þ
a!n
where b Ya ðc; a; xÞ ¼
eax xncþb Cð1 c þ bÞ 1 F1 ðn; 1 þ n c þ b; axÞ Cð1 þ n c þ bÞ
Proof. Taking the limits as a ! n ; nþ respectively we get the result.
Corollary 4.4. If a; b 2 R, then lim Lba ðc; a; xÞ ¼ Lbn ðc; a; xÞ ¼ limþ Lba ðc; a; xÞ
a!n
a!n
ð30Þ
Lemma 4.5. Let a; c 2 ð0; 1Þ and b; a 2 R, then lim Lba ðc; a; xÞ ¼ xc ¼ lim Lba ðc; a; xÞ
a!0þ
a!0
ð31Þ
38
S.Z. Rida, A.M.A. El-Sayed / Appl. Math. Comput. 147 (2004) 29–43
and lim Lba ðc; a; xÞ ¼ xc ¼ lim Lba ðc; a; xÞ
a!0þ
ð32Þ
a!0
Corollary 4.6. If a ¼ c, a ¼ 1, then lim Lba ða; 1; xÞ ¼ lim Lba ða; 1; xÞ ¼ 1 ¼ limþ Lba ða; 1; xÞ ¼ lim Lba ða; 1; xÞ
a!0þ
a!0
a!0
a!0
ð33Þ Let us first recall some known facts. The abstract Bell polynomials Bn ¼ Bn ðx1 ; x2 ; . . . ; xn Þ (n ¼ 1; 2; . . .) in variables x1 ; x2 ; . . . ; xn are given by the recursion Bnþ1 ¼
n X n k¼0
k
Bk xnþ1k
for n ¼ 0; 1; . . . together with B0 ¼ 1, while the nth Bell differential polynomials, is the result of n times applications of D þ u, D ¼ d=dx to the constant function 1: Bn ðu; u0 ; . . . ; uðn1Þ Þ ¼ ðD þ uÞn 1 The two definitions are equivalent to each other, cf. [9–11], and are valid in the matrix case too, where x1 ; x2 ; . . . ; xn or u ¼ uðxÞ respectively have values in an algebra of matrices. Now, we define the function Yab ðc; a; xÞ by the use of the Bell polynomials Bn ðxÞ, n ¼ 0; 1; 2; . . . (see [9–11]). Definition 4.7. Let a; c 2 ðn 1; nÞ, n ¼ 1; 2; . . . and b; a 2 R; a > 0. The function Yab ðc; a; xÞ of order a is defined by Yab ðc; a; xÞ
¼I
na
n X n k¼0
k
Cð1 þ c þ bÞ eax Bk ða; 0; . . . ; 0Þxcnþbþk Cð1 þ c n þ b þ kÞ ð34Þ
By this new definition, we can prove the following lemma Lemma 4.8. Let a; c 2 ðn 1; nÞ, n ¼ 1; 2; . . . and b; a 2 R; a > 0. Then lim Yab ðc; a; xÞ ¼ Ynb ðc; a; xÞ a!n
ð35Þ
S.Z. Rida, A.M.A. El-Sayed / Appl. Math. Comput. 147 (2004) 29–43
39
Proof. ( lim Yab ðc; a; xÞ a!n
¼ lim I
n X n
na
a!n
k
k¼0
Cð1 þ c þ bÞ Cð1 þ c n þ b þ kÞ )
eax Bk ð a; 0; . . . ; 0Þxcnþbþk ( ¼ lim /na ðxÞ a!n
n X n k¼0
k
Cð1 þ c þ bÞ Cð1 þ c n þ b þ kÞ )
eax Bk ð a; 0; . . . ; 0Þxcnþbþk ¼ /0 ðxÞ
n X n
k
k¼0
Cð1 þ c þ bÞ Cð1 þ c n þ b þ kÞ
ax
¼
e Bk ða; 0; . . . ; 0Þxcnþbþk n X n Cð1 þ c þ bÞ k¼0
k
Cð1 þ c n þ b þ kÞ
eax Bk ða; 0; . . . ; 0Þxcnþbþk ¼ Ynb ðc; a; xÞ
5. Recurrence relations and the differential equation We now give some recurrence relations for the functions Yab ðc; a; xÞ, Lba ðc; a; xÞ and we prove that these functions are particular solutions of second order differential equations, respectively xD2 Yab ðc; a; xÞ þ ð1 þ a c b þ axÞDYab ðc; a; xÞ þ ð1 þ aÞaYab ðc; a; xÞ ¼ 0 xD2 Lba ðc; a; xÞ þ ð1 þ a þ b c axÞDLba ðc; a; xÞ ða cÞb b La ðc; a; xÞ ¼ 0 þ ac þ x
Theorem 5.1. Let a; c 2 ðn 1; nÞ, n ¼ 1; 2; . . . and b; a 2 R; a > 0. Then b1 b Yab ðc; a; xÞ ¼ ðc þ bÞYa1 ðc; a; xÞ aYa1 ðc; a; xÞ b b ¼ ðc þ bÞYa1 ðc 1; a; xÞ aYa1 ðc; a; xÞ
ð36Þ
40
S.Z. Rida, A.M.A. El-Sayed / Appl. Math. Comput. 147 (2004) 29–43 b DYab ðc; a; xÞ ¼ Yaþ1 ðc; a; xÞ ¼ ðc þ bÞYab1 ðc; a; xÞ aYab ðc; a; xÞ b ¼ ðc þ bÞDYa1 ðc 1; a; xÞ aYab ðc; a; xÞ
ð37Þ
b ðc; a; xÞ Dn Yab ðc; a; xÞ ¼ Yaþn n X n Cð1 þ c þ bÞ nk Yabk ðc; a; xÞ ¼ ðaÞ k Cð1 þ c þ b kÞ k¼0
ð38Þ
b b xDYa1 ðc 1; a; xÞ Yab ðc; a; xÞ ¼ aYa1 ðc 1; a; xÞ
ð39Þ
Proof. From the definition of Yab ðc; a; xÞ, we have Yab ðc; a; xÞ ¼ Da1 ðc þ bÞxcþb1 eax axcþb eax ¼ ðc þ bÞDa1 xðc1Þþb eax aDa1 xcþb eax b1 b ¼ ðc þ bÞYa1 ðc; a; xÞ aYa1 ðc; a; xÞ
Also, DYab ðc; a; xÞ ¼ Da ðc þ bÞxcþb1 eax axcþb eax ¼ ðc þ bÞYab1 ðc; a; xÞ aYab ðc; a; xÞ b b ¼ ðc þ bÞDYa1 ðc 1; a; xÞ aYa1 ðc; a; xÞ
where b1 DYa1 ðc; a; xÞ ¼ Yab1 ðc; a; xÞ
and also n
D
Yab ðc; a; xÞ
n X n ðDk xcþb ÞðDnk eax Þ ¼D k k¼0 n X Cð1 þ c þ bÞ n nk Da xcþbk eax ðaÞ ¼ k Cð1 þ c þ b kÞ k¼0 n X Cð1 þ c þ bÞ n Y bk ðc; a; xÞ ¼ ðaÞnk k Cð1 þ c þ b kÞ a k¼0 a
From the convergence of the power series expansion of eax xcþb and the properties of the fractional derivative we obtain Yab ðc; a; xÞ ¼
m 1 X ðaÞ Cð1 þ m þ c þ bÞ mþcaþb x Cð1 þ m þ c a þ bÞ m! m¼0
from which we can prove (by direct substitution) the last result.
ð40Þ
S.Z. Rida, A.M.A. El-Sayed / Appl. Math. Comput. 147 (2004) 29–43
41
Theorem 5.2. Let a; c 2 ðn 1; nÞ, n ¼ 1; 2; . . . and b; a 2 R; a > 0. Then b xDLba ðc; a; xÞ ¼ ðc þ bÞLb1 a ðc; a; xÞ þ ð2ax bÞLa ðc; a; xÞ
þ abLba ðc; a; xÞ ¼ ðc þ bÞ axÞLba1 ðc 1; a; xÞ þ xDLba1 ðc
ð41Þ
axDLba ðc; a; xÞ
1; a; xÞ n Cða þ 1Þ X n k bk k! x Lk ðxÞ½Dk Lba ðc; a; xÞ Lbaþn ðc; a; xÞ ¼ k Cða þ n þ 1Þ k¼0 ½ðb
ð42Þ ð43Þ
where Lbk k ðxÞ is the wellknown Laguerre polynomials. xDLba1 ðc 1; a; xÞ aLba ðc; a; xÞ ¼ ðax b aÞLba1 ðc 1; a; xÞ
ð44Þ
Proof. From the definition of Lba ðc; a; xÞ and Eqs. (36)–(39) respectively, we have the result. Theorem 5.3. The function Yab ðc; a; xÞ is particular solution of the differential equation xD2 Yab ðc; a; xÞ þ ð1 þ a c b þ axÞDYab ðc; a; xÞ þ ð1 þ aÞaYab ðc; a; xÞ ¼ 0
ð45Þ
Proof. Multiplying (37) by x and from (39) into it we have xDYab ðc; a; xÞ ¼ axYab ðc; a; xÞ þ ðc þ bÞYab ðc; a; xÞ b ðc 1; a; xÞ aðc þ bÞYa1
ð46Þ
Differentiating this equation and from (37) into it, we obtain the result.
Theorem 5.4. The function Lba ðc; a; xÞ is particular solution of the differential equation xD2 Lba ðc; a; xÞ þ ð1 þ a þ b c axÞDLba ðc; a; xÞ ða cÞb b La ðc; a; xÞ ¼ 0 þ ac þ x Proof. From (3) into (45), we obtain the result.
ð47Þ
6. Orthogonality property Theorem 6.1. For any real numbers a1 6¼ a2 , c1 6¼ c2 , we have Z 1 eax xb Lba1 ðc1 ; a; xÞLba2 ðc2 ; a; xÞ dx ¼ 0 0
ð48Þ
42
S.Z. Rida, A.M.A. El-Sayed / Appl. Math. Comput. 147 (2004) 29–43
for a1 ¼ c1 , a2 ¼ c2 . Proof. Let uba ðc; a; xÞ ¼ eax=2 xb=2 Lba ðc; a; xÞ, a 2 Rþ , then by direct calculation we can prove that xD2 uba ðc; a; xÞ þ ð1 þ a cÞDuba ðc; a; xÞ a b b2 a2 þ ð1 þ a þ c þ bÞ þ ða cÞ x uba ðc; a; xÞ ¼ 0 2 2x 4x 4
ð49Þ
Then for any positive real numbers a1 ; c1 and a2 ; c2 , we have xD2 uba1 ðc1 ; a; xÞ þ ð1 þ a1 c1 ÞDuba1 ðc1 ; a; xÞ a b b2 a2 þ ð1 þ a1 þ c1 þ bÞ þ ða1 c1 Þ x uba1 ðc1 ; a; xÞ ¼ 0 ð50Þ 2 2x 4x 4 xD2 uba2 ðc2 ; a; xÞ þ ð1 þ a2 c2 ÞDuba2 ðc2 ; a; xÞ a b b2 a2 þ ð1 þ a2 þ c2 þ bÞ þ ða2 c2 Þ x uba2 ðc2 ; a; xÞ ¼ 0 ð51Þ 2 2x 4x 4 By multiplying the first equation by uba2 ðc2 ; a; xÞ and the second equation by uba1 ðc1 ; a; xÞ, subtracting the resulting equations and integrating from 0 to 1 we obtain Z 1 Z 1 uba1 ðc1 ; a; xÞuba2 ðc2 ; a; xÞ dx ¼ ðc1 a1 Þ uba2 ðc2 ; a; xÞDuba1 ðc1 ; a; xÞ dx 0 0 Z 1 þ ða2 c2 Þ uba1 ðc1 ; a; xÞDuba2 ðc2 ; a; xÞ dx 0
ð52Þ Then, from which, for a1 ¼ c1 and a2 ¼ c2 , we have the result.
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[7] I. Podlubny, A.M.A. El-Sayed, On two definitions of fractional calculus, Slovak Academy of Sciences Institute of Experimental Physics, UEF-03-96, ISBN 80-7099-252-2, 1996. [8] I. Podlubny, Fractional Differential Equation, Academic Press, San Diego, 1999. [9] R. Schimming, S.Z. Rida, Non-commutative Bell polynomials, Int. J. Algebra Comput. 6 (1996) 635–644. [10] R. Schimming, S.Z. Rida, The Bell differential polynomials, in: G.E. Bergum, et al. (Eds.), Applications of Fibonacci Numbers, vol. 7, Kluwer, Dordrecht, 1998, pp. 353–367. [11] S.Z. Rida, Powers of differential operators and explicit differential polynomials, Ph.D. Thesis, Free University Berlin, ISBN 3-8265-1670-2, 1996.