American Journal of M athematics and Statistics 2012, 2(6): 221-225 DOI: 10.5923/j.ajms.20120206.10
On Applications of Fractional Calculus Involving Summations of Series Praveen Agarwal Department of M athematic Anand International College of Engineering Jaipur, 302012, India
Abstract A significantly large nu mber of earlier works on the subject of fractional calculus give interesting account of the
theory and applications of fractional calculus operators in many different areas of mathematical analysis (such as ordinary and partial differential equations, integral equations, special functions, summation of series, et cetera). The main object of the present paper is to obtain number of summat ions of series concerning generalized hypergeometric functions. Our finding provides interesting unifications and extensions of a number of new and known results.
Keywords
Fractional Calcu lus, Special Function, Su mmat ion of Series, Generalized Leibniz Ru le, Generalized Hypergeometric Series, Laguerre Po lynomials
1. Introduction One of the most frequently encountered tools in the theory of fractional calcu lus (that is, differentiat ion and integration of an arbitrary real or co mplex order) is furn ished by the familiar
differintegral operator
a Dz
α
defined
and
represented by Oldham and Spanier[12]:
a Dz
α
f (z) =
z 1 ( z − y ) f ( y ) dy ) Γ ( −α ) ∫a (a ∈ ; ℜ(α ) < 0 −α −1
(1.1)
and z 1 n −α −1 D z Γ ( n − α ) ∫ ( z − y ) f ( y ) dy a (1.2) α −n n = D z D f ( z ) , ℜ(α ) ≥ 0. a z
f z = a Dz ( ) α
n
corresponding essentially to the classical Riemann-Liouville fractional derivative (or integral) of order α (or – α ). Moreover, when a → ∞ , Equation (1.1) may be identified with the definition of the familiar Weyl fract ional derivative (or integral) o f order α (or – α ). In recent years there has appeared a great deal of literature discussing the application of the aforementioned fractional calculus operators in a number of areas of mathematical analysis (such as ordinary and partial differential equations, integral equations, summation of series, et cetera) and now stands on fairly firm footing through the research contribution of various authors (cf., e.g.,[2],[5-7],[9-14],[16] and[17]). In the present paper main object is to obtain number of summations of series concerning generalized hypergeometric functions. The familiar Leibniz rule for ord inary derivatives admits itself of the fo llo wing extension in terms of the
where n is the least positive integer such that n>q. {The operator
and
D −1 }.
For a=0 the operator
D
α z
is given by
z
defined by (1.3):
∞ α u z v z = ( ) ( ) Dz ∑ n n =0
a D z provides a generalization of the
d D≡ dz
α
α
α
familiar differential and integral operator, v iz.,
D
Riemann-Liouville operator
D
α −n z
(1.4)
u ( z ) D z v ( z ) (α ∈ ) n
The generalized Leibniz rule (1.4), wh ich was also applied earlier by Galué et al.[5] o rder to derive the summation identity:
(α ) n ∞ α − k −n n+ k ∑ k D z u ( z ) D z v ( z ) (1.3) = α ∈ ( ) ! n (1.5) =n 1 =k 0 z 0 z ∞ α −n n = ∑ n D z u ( z ) D z v ( z ) (α ∈ ) , * Corresponding author: n =1
[email protected] (Praveen Agarwal) Suffers fro m an apparent drawback in the sence that the Published online at http://journal.sapub.org/ajms interchange of the function u(z) and v(z) on the right-hand Copyright © 2012 Scientific & Academic Publishing. All Rights Reserved
D
α
D
α
∞
∑ ( −1)
n −1
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Praveen A garwal: On Applications of Fractional Calculus Involving Summations of Series
side is not obvious. (see also Galué et al. for several symmetrical generalized of (1.4) considered by Watanabe[17] summation formu las[6] contained in the Chen-Srivastava[2] ) and Osler[13], without such a drawback, is given by (cf., e.g., which she deduced by suitable specializing the function u(z) Samko et al.[14, p. 316, Equation (17.12)]): and v(z) in the summation identity (1.5) above.) A further
which, in the special case when
α
∞
∑ η + n D
D z u ( z ) v ( z ) α
n = −∞
(η = 0 )
α −η − n z
u ( z ) D z v ( z ) η +n
( α ,η ∈ )
(1.6)
,yields the Leibniz rule (1.4).
The condition of valid ity of the above results is given by T. J. Osler[13, p. 664-665]). The generalized hypergeometric function of one variable viz., p.19]) is also required here:
p
Fq [.;.; z ] defined and represented as follows (see e.g.[15,
p
( a p ) ; ∞ ∏ ( a j ) n z n j =1 z = ; provided ∑ p Fq n! ( bq ) ; n =0 q ∏ ( b j )n
p ≤ q or
p= q + 1 and
z 1, ℜ(λ ) > 0, and
ℜ( µ ) > 0.
The conditions of validity of the above results follow easily fro m the conditions given by T. J. Osler[13, p. 664-665]). Summati on Formulae 2.2 ∞
α 1 F1 ( µ ; µ − α + η + n; iaz ) − 1 F1 ( µ ; µ − α + η + n; −iaz ) Γ ( λ −η − n ) Γ ( µ − α + η + n )
∑ n = −∞ η + n
1 F1 ( λ ; λ − η − n; ibz ) + 1 F1 ( λ ; λ − η − n; −ibz ) (2.2) ( 2i ) Γ ( µ + λ − 1) [ 1 F1 ( µ + λ − 1; µ + λ − α − 1; i ( a + b ) z ) − 1 F1 ( µ + λ − 1; µ + λ − α − 1; −i ( a + b ) z ) Γ ( µ ) Γ ( λ ) Γ ( µ + λ − α − 1) + 1 F1 ( µ + λ − 1; µ + λ − α − 1; i ( a − b ) z ) − 1 F1 ( µ + λ − 1; µ + λ − α − 1; −i ( a − b ) z ) provided that
ℜ ( λ + µ ) > 1, ℜ(λ ) > 0, and
ℜ( µ ) > 0.
American Journal of M athematics and Statistics 2012, 2(6): 221-225
223
The conditions of validity of the above results follow easily fro m the conditions given by T. J. Osler[13, p. 664-665]). Summati on Formulae 2.3 ∞
α 2 F2 ( a + m + 1, µ ; a + 1, µ − α + η + n; −k1 z ) 1 F1 ( λ ; λ − η − n; − k 2 z ) Γ ( λ −η − n ) Γ ( µ − α + η + n )
∑ n =−∞ η + n
Γ ( µ + λ − 1)
Γ ( µ ) Γ ( λ ) Γ ( µ + λ − α − 1)
where
ℜ ( λ + µ ) > 1, ℜ(λ ) > 0, and
2
F2 ( a + m + 1, µ + λ − 1; a + 1, µ + λ − α − 1; − ( k1 + k2 ) z )
(2.3)
ℜ( µ ) > 0.
The conditions of validity of the above results follow easily fro m the conditions given by T. J. Osler[13, p. 664-665]). Summati on Formulae 2.4
1 −a 2 z µ µ α η F n ; , ; − + + 1 1 ∞ 4 1 α −b 2 z 2 − − λ λ η F n ; , ; ∑ 1 1 4 n =−∞ η + n Γ ( λ − η − n ) Γ ( µ − α + η + n ) 2 2 a + b) z ( 1 1 F1 µ + λ − 1; , µ + λ − α − 1; − = Γ ( µ + λ − α − 1) 2 4 2 a − b ) z ( 1 + 1 F1 µ + λ − 1; , µ + λ − α − 1; − 2 4
Γ ( µ + λ − 1)
provided that
ℜ ( λ + µ ) > 1, ℜ(λ ) > 0, and
(2.4)
ℜ( µ ) > 0.
The conditions of validity of the above results follow easily fro m the conditions given by T. J. Osler[ 13, p. 664-665]). Summati on Formulae 2.5
Γ(µ ) α a 2 z 1/ 2 1 − + + µ µ α η F n ; , ; ∑ z 1 2 2 4 n = −∞ η + n ( µ − α +η + n) ∞
+
1 aΓ µ + 2
1 b 2 z 1/2 1 1 1 a2 z Γ ( λ ) F n F n µ µ α η λ λ η ; , ; ; , ; + − + + + × − − z 1 2 1 1 2 2 2 2 4 ( λ − η − n ) 2 4 µ − α +η + n + 2 1 bΓ λ + 2 b z 1 1 1 2 F λ + ; , λ −η − n + ; + 1 1 2 2 2 2 4 (2.5) λ −η − n + 2 2 Γ ( µ + λ − 1) ( a + b ) z z 1 F µ λ µ λ α 1; , 1 ; + − + − − 1 1 2 4 ( µ + λ − α − 1)
+
where
1 2 a b z + ) z1/2 1 3 1 ( 2 ; , µ + λ −α − ; 1 F2 µ + λ − 1 2 2 2 4 µ + λ −α − 2
( a + b ) Γ µ + λ −
ℜ ( λ + µ ) > 1, ℜ(λ ) > 0, and
ℜ( µ ) > 0.
The conditions of validity of the above results follow easily fro m the conditions given by T. J. Osler[ 13, p. 664-665]). Proofs: The results are obtained by assigning particular values to the functions u (z) and v (z) in the generalized Leibniz ru le (1.6).
224
Praveen A garwal: On Applications of Fractional Calculus Involving Summations of Series
µ −1 = cos az and v ( z ) z λ −1 cos bz If we put u ( z ) z=
u ( z ) v ( z ) D D= α
α
z
z
in (1.6), then L.H.S. of (1.6) beco mes
1 µ +λ −2 {cos ( a + b ) z + cos ( a − b ) z} 2 z
and using known result[4, p.189, eqn. (32)], we get
1 Γ ( µ + λ − 1) [ 1 F1 ( µ + λ − 1; µ + λ − α − 1; i ( a + b ) z ) + 1 F1 ( µ + λ − 1; µ + λ − α − 1; −i ( a + b ) z ) 4 Γ ( µ + λ − α − 1) F ( µ + λ − 1; µ + λ − α − 1; i ( a − b ) z ) + 1 F1 ( µ + λ − 1; µ + λ − α − 1; −i ( a − b ) z ) z
1 1
(2.6)
µ + λ −α − 2
For R.H.S., we similarly have
∞ α α −η − n α α −η − n µ −1 η +n z cos az Dηz + n z λ −1 cos bz , (2.7) u z v z = ( ) ( ) ∑ ∑ D z Dz D z n = −∞ η + n n = −∞ η + n Γ(µ ) 1 F1 ( µ ; µ − α + η + n; iaz ) + 1 F1 ( µ ; µ − α + η + n; −iaz ) z µ −α +η + n −1 = Dzα −η − n z µ −1 cos az 2Γ ( µ − α + η + n ) ∞
Γ (λ ) 1 F1 ( λ ; λ − η − n; ibz ) + 1 F1 ( λ ; λ − η − n; −ibz ) z λ −η − n −1 = Dηz + n z λ −1 cos bz 2Γ ( λ − η − n ) putting (2.6) and (2.7), in (1.6), we have the required result (2.1) after a little simp lification:
µ −1 Again, if we put u ( z ) z= sin az and v( z ) z λ −1 cos bz in (1.6), proceed on similar lines as adopted in (2.1) =
and using known results[4, p.188, Eq. (21)], we obtained the required interesting formulae (2.2). Next, If we take
u( z) = z µ −1 Lam ( ( k1 + k2 ) z ) e − k1z
and
v( z ) = z λ −1e − k2 z in (1.6), proceed on similar lines as
adopted in (2.1) and using known results[4, p.193, Eq. (51) and p.187, Eq. (14)], we arrive at the required interesting formu lae (2.3).
(
)
(
µ −1 Further, on putting u ( z ) z= = cos az1/2 and v( z ) z λ −1 cos bz1/2
)
in (1.6), we easily obtained the
formulae (2.4) after a litt le simp lification on making use of similar lines of proof as adopted in (2.1) and using known results[4, p.190, Eq. (35)]:
(
)
(
)
µ −1 Similarly, if we take u ( z ) z= = exp az1/2 and v( z ) z λ −1 exp bz1/2 in (1.6), we easily arrive at the
required formu lae (2.5) after a little simp lificat ion on making use of similar lines of proof as adopted in (2.1) and using known results[4, p.190, Eq. (35)]
3. Special Cases In view of the large number of parameters involved in the summat ions of series established above, these summations of series are capable of yielding a number of known and new results. We record here only one special case for lack of space. For example: If, we take a = b = 0; µ = µ − c; λ = λ − d ; α = λ − c − 1 and η = λ − 1 in (2.1) and making use of the following well-known result on both the sides of the resulting result of (2.1) (cf., e.g., Erdély i et al.[4, p.185, Eqn. 13.1 (7)]):
= Dz−α { z λ }
Γ(λ + 1) λ +α z (ℜ(α ) > 0; ℜ(λ ) > −1), We easily arrive at the well-known Dougall’s formu la[3, p. Γ(λ + α + 1)
7, Eqn. 1.4(1)] after a little simp lification.
ACKNOWLEDGEMENTS
REFERENCES
The authors are thankful to the referee for the valuable comments and suggestions, which have led the paper to the present form.
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