On the Riemann–Liouville fractional calculus, g-Jacobi functions and F-Gauss functions

Applied Mathematics and Computation 187 (2007) 315–325 www.elsevier.com/locate/amc

On the Riemann–Liouville fractional calculus, g-Jacobi functions and F-Gauss functions S.P. Mirevski *, L. Boyadjiev, R. Scherer Institute of Practical Mathematics, University of Karlsruhe, 76128 Karlsruhe, Germany

Dedicated to Professor H.M. Srivastava on the occasion of his 65th birthday

Abstract This paper refers to a fractional extension of the classical Jacobi polynomials. A fractional order Rodrigues’ type representation formula is considered. By means of the Riemann–Liouville operator of fractional calculus, new g-Jacobi functions are defined, some of their properties are given and compared with the corresponding properties of the classical Jacobi polynomials. Furthermore, the hypergeometric equation of Gauss is extended to a fractional order. A new F-Gauss hypergeometric function is defined as a solution to the extended fractional differential equation and considered as a candidate for a fractional hypergeometric function of Gauss. Ó 2007 Elsevier Inc. All rights reserved. Keywords: Riemann–Liouville fractional differentiation and integration operators; Jacobi polynomials; Rodrigues’ representation; g-Jacobi functions; Gauss hypergeometric differential equation; F-Gauss hypergeometric functions

1. Introduction Fractional calculus is ‘‘the theory of integrals and derivatives of arbitrary order, which unify and generalize the notions of integer-order differentiation and n-fold integration’’ [8]. The idea of generalizing differential operators to a non-integer order, in particular to the order 12, first appears in the correspondence of Leibniz with L’Hoˆpital (1695), Johann Bernoulli (1695), and John Wallis (1697) as a mere question or maybe even play of thoughts. In the following three hundred years a lot of mathematicians contribute to the fractional calculus: Laplace (1812), Lacroix (1812), Fourier (1822), Abel (1823–1826), Liouville (1832–1837), Riemann (1847), Gru¨nwald (1867–1872), Letnikov (1868–1872), Sonin (1869), Laurent (1884), Heaviside (1892– 1912), Weyl (1917), Davis (1936), Erde´lyi (1939–1965), Gel’fand and Shilov (1959–1964), Dzherbashian (1966), Caputo (1969), and many others [4–7,9,10].

*

Corresponding author. E-mail addresses: [email protected] (S.P. Mirevski), [email protected] (L. Boyadjiev), scherer@ math.uni-karlsruhe.de (R. Scherer). 0096-3003/$ - see front matter Ó 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2007.01.035

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Yet, it is only after the First Conference on Fractional Calculus and its Applications that the fractional calculus becomes one of the most intensively developing areas of mathematical analysis. Its fields of application range from biology through physics and electrochemistry to economics, probability theory and statistics. On behalf of the nature of their definition the fractional derivatives provide an excellent instrument for the modeling of memory and hereditary properties of various materials and processes. Half-order derivatives and integrals prove to be more useful for the formulation of certain electrochemical problems than the classical methods [3]. Fractional differentiation and integration operators are also used for extensions of the diffusion and wave equations [11] and, recently, of the temperature field problem in oil strata [1]. In this paper we introduce some of the basic properties of the Riemann–Liouville operators of fractional calculus. The Rodrigues’ type representation formula for the classical Jacobi polynomials is generalized by means of the Riemann–Liouville fractional differentiation operator. The so-called g-Jacobi functions are defined and some of their properties are studied. Again by means of the Riemann–Liouville fractional operators, the hypergeometric equation of Gauss is extended to a fractional order. A solution of the extended fractional differential equation is obtained by a modified power series method. This solution has the Gauss hypergeometric function as a particular case and, therefore, is considered as a candidate for a fractional hypergeometric function of Gauss. 2. Basic properties of the fractional operators Definition 1. For t > 0, Z t 1 m1 J m f ðtÞ :¼ ðt  sÞ f ðsÞ ds CðmÞ 0

ð1Þ

is called the Riemann–Liouville fractional integral of the function f ðtÞ of order m with ReðmÞ > 0 [6,8]. According to the literature [4,6,8] the following properties of the Riemann–Liouville fractional integrals hold. Theorem 2. Let f ðtÞ and gðtÞ be such that both J m f ðtÞ and J m gðtÞ exist. Then, the following basic properties of the Riemann–Liouville integrals hold: (i) interpolation (continuity) lim J m f ðtÞ ¼ J n ðtÞ; m!n

where J n (n 2 N) is the classical operator for n-fold integration; (ii) linearity J m ½kf ðtÞ þ gðtÞ ¼ kJ m f ðtÞ þ J m gðtÞ;

ðk 2 CÞ;

(iii) semi-group property (law of exponents) J l ½J m f ðtÞ ¼ J lþm f ðtÞ; (iv) commutativity J lJ m ¼ J mJ l: Definition 3. If t > 0 and m 2 N such that m  1 6 l < m, then the fractional derivative of f ðtÞ of order l is defined as Dl f ðtÞ ¼ Dm ½J ml f ðtÞ (if it exists) where m  l > 0 [6,8]. The following properties of the Riemann–Liouville fractional derivatives hold [4,6,8].

ð2Þ

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317

Theorem 4. Let f ðtÞ and gðtÞ be such that both Dl f ðtÞ and Dl gðtÞ exist. Then, the following basic properties of the Riemann–Liouville derivatives hold: (i) interpolation (continuity) lim Dl f ðtÞ ¼ f ðmÞ ðtÞ;

l!m

(ii) linearity Dl ½kf ðtÞ þ gðtÞ ¼ kDl f ðtÞ þ Dl gðtÞ;

ðk 2 CÞ;

(iii) the semi-group property (law of exponents) does not hold, i.e., in general Dl ½Dm f ðtÞ 6¼ Dlþm f ðtÞ; (iv) non-commutativity: in general Dl Dm 6¼ Dm Dl : The interplay between differentiation and integration has its analogy for their fractional counterparts as well. The following proposition holds [8, pp. 70–71]. Theorem 5. For m > 0, t > 0, n  1 6 m < n (n 2 N) (i) Dm ½J m f ðtÞ ¼ f ðtÞ; (ii) If the fractional derivative Dm f ðtÞ of f ðtÞ is integrable, then J m ½Dm f ðtÞ ¼ f ðtÞ 

n X k¼1

½Dmk f ðtÞt¼0

tmk : Cðm  k þ 1Þ

Central for our considerations in Section 4 is the following result [8, pp. 91–97]. Theorem 6 (Leibniz rule for fractional differentiation). If f ðsÞ is continuous in ½0; t and uðsÞ has n þ 1 continuous derivatives in ½0; t, then the fractional derivative of the product uðtÞf ðtÞ (for l > 0) is given by the Leibniz rule for fractional differentiation m   X l Dl ½uðtÞf ðtÞ ¼ uðkÞ ðtÞDlk f ðtÞ  Rlm ðtÞ; k k¼0 where l > 0, m P l þ 1 and Z t Z t 1 l1 n ðt  sÞ f ðsÞ uðmþ1Þ ðnÞðt  nÞ dn ds: Rlm ðtÞ ¼ m!CðlÞ 0 s If, in addition, uðsÞ along with all its derivatives is continuous in ½0; t, the Leibniz rule takes the form 1   X l l D ½uðtÞf ðtÞ ¼ uðkÞ ðtÞDlk f ðtÞ: k k¼0

ð3Þ

3. Examples In this section two simple examples of a fractional integral and derivative are given. Proposition 7. The fractional integral of the constant function is given by J m1 ¼

1 tm : Cðm þ 1Þ

ð4Þ

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Proof. Showing (4) is straightforward. By definition (1), J m1 ¼

1 CðmÞ

Z

t

ðt  sÞ

m1

ds ¼ 

0

1 tm m s¼t ; ½ðt  sÞ s¼0 ¼ mCðmÞ mCðmÞ

and property zCðzÞ ¼ Cðz þ 1Þ

ð5Þ

of the gamma function yields (4). h Fig. 1 illustrates three special cases of the fractional integral of f ðtÞ ¼ 1. Fig. 2 shows its three-dimensional graph, where integrals of order 0 6 l 6 2 on the interval ½0; 4 are considered. Proposition 8 (Fractional derivative of the power function). If l P 0, t > 0 and a > 1, then Dl t a ¼

Cða þ 1Þ al t : Cða  l þ 1Þ

ð6Þ

Proof. Letting m  1 6 l < m, m 2 Nþ , from (2) we get   Z t dm 1 ml1 a ðt  sÞ s ds : D t ¼ m dt Cðm  lÞ 0 l a

Substituting s ¼ tk yields

d l ta ¼

2

3

Z 1 7 1 dm 6 6 mþal 7 ml1 a t k ð1  kÞ dk 6 7: 5 Cðm  lÞ dtm 4 0 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} Bðaþ1;mlÞ

Using the property Bða þ 1; m  lÞ ¼

Cða þ 1ÞCðm  lÞ Cðm þ a  l þ 1Þ

4 0 – integral 1/3–integral 1/2–integral 3/4–integral 1 – integral

3.5 3

2.5 2 1.5 1 0.5 0

0

0.5

1

1.5

2

2.5

3

Fig. 1. Fractional integrals of f ðtÞ ¼ 1.

3.5

4

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319

8

ν

J (1)

6

4

2

0 2 1.5

4 3

1 2

0.5 nu

1 0

t

0

Fig. 2. Fractional integrals of f ðtÞ ¼ 1 of order 0 6 m 6 2 on the interval ½0; 4.

of the beta function and m times integer-order differentiation, we get Dl ta ¼

Cða þ 1Þ ðm þ a  lÞðm þ a  l  1Þ    ða  l þ 1Þtal : Cðm þ a  l þ 1Þ

Finally, applying m times (5) produces the desired result.

h

Using the same idea, it can be shown [6,8] that formula (6) holds for negative values of l as well. In this case, Dl is to be considered as J l . The three-dimensional graph of the fractional derivatives of f ðtÞ ¼ t2 of order 0 6 l 6 2 on the interval ½0; 4 can be seen in Fig. 3.

20

μ

D t2

15

10

5

0 2 1.5

4 3

1 mu

2

0.5

1 0

0

t

Fig. 3. Fractional derivatives of f ðtÞ ¼ t2 of order 0 6 l 6 2 on the interval ½0; 4.

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4. g-Jacobi functions The classical Jacobi polynomials are usually defined by the Rodrigues’ type formula 1

n

a

P nða;bÞ ðxÞ ¼ ð2Þ ðn!Þ ð1  xÞ ð1 þ xÞ

b

dn nþa nþb ½ð1  xÞ ð1 þ xÞ ; dxn

ð7Þ

where (for integrability purposes) a > 1, b > 1 [2]. Their basic properties [2,12] are given in Table 1. Our idea is to extend the Jacobi polynomials by taking the Riemann–Liouville fractional derivative (2) in (7) and substituting m 2 R for n 2 N. Definition 9. We define the (generalized or) g-Jacobi functions by the formula m 1 a b m mþa P a;b ð1 þ tÞmþb ; m > 0; m ðtÞ ¼ ð2Þ Cðm þ 1Þ ð1  tÞ ð1 þ tÞ D ½ð1  tÞ

ð8Þ

where a > 1, b > 1 and Dm is the Riemann–Liouville fractional differentiation operator (2). In the following we derive some properties of the g-Jacobi functions, analogous to the properties of the classical Jacobi polynomials. Theorem 10 (Explicit formula). For the g-Jacobi functions holds the explicit formula   1  X mþa mþb k mk m ða;bÞ P m ðtÞ ¼ 2 ðt  1Þ ðt þ 1Þ ; mk k k¼0

ð9Þ

where   a Cð1 þ aÞ ¼ Cð1 þ bÞCð1 þ a  bÞ b

ð10Þ

is the binomial coefficient with real arguments. mþb

Proof. Applying the Leibniz rule (3) (since ð1 þ sÞ is continuous along with all its derivatives in ½0; t) to the fractional derivative in (8) yields 1   X m ð2Þm a b mþb mþa ða;bÞ ð1  tÞ ð1 þ tÞ P m ðtÞ ¼ fDk ½ð1 þ tÞ gfDmk ½ð1  tÞ g: Cðm þ 1Þ k k¼0 Using the generalized binomial theorem 1   X a k ak xy ðx þ yÞa ¼ k k¼0

ð11Þ

Table 1 Properties of the classical Jacobi polynomials and the g-Jacobi functions Property

g-Jacobi functions

Definition

P a;b m ðtÞ

m

Jacobi polynomials 1

a

b

Equation

¼ ð2Þ Cðm þ 1Þ ð1  tÞ ð1 þ tÞ Dm ½ð1  tÞmþa ð1 þ tÞmþb     1 P mþa mþb P ða;bÞ ðt  1Þk ðt þ 1Þmk ðtÞ ¼ 2m m mk k k¼0   mþa P ða;bÞ ðtÞ ¼ F 1 ðm; m þ a þ b þ 1; a þ 1; 1t m 2 Þ m  2 mþa P ða;bÞ ð1Þ ¼ m m   m þb P ða;bÞ ð1Þ ¼ m m ð1  t2 Þy 00 þ ½b  a  ða þ b þ 2Þty 0 þ mðm þ a þ b þ 1Þy ¼ 0

Derivative

d ða;bÞ ðtÞ dt P m

Explicitly Explicitly Value at 1 Value at 1

ðaþ1;bþ1Þ

¼ 12 ðm þ a þ b þ 1ÞP m1

ðtÞ

P na;b ðtÞ ¼ ð2Þn ðn!Þ1 ð1  tÞa ð1 þ tÞb n  dtd n ½ð1  tÞnþa ð1 þ tÞnþb     1 P nþa nþb ðt  1Þk ðt þ 1Þnk P nða;bÞ ðtÞ ¼ 2n nk k k¼0   nþa P nða;bÞ ðtÞ ¼ F 1 ðn; n þ a þ b þ 1; a þ 1; 1t 2 Þ n  2 nþa P nða;bÞ ð1Þ ¼ n   n þb P nða;bÞ ð1Þ ¼ n ð1  t2 Þy 00 þ ½b  a  ða þ b þ 2Þty 0 þ nðn þ a þ b þ 1Þy ¼ 0 d ða;bÞ ðtÞ dt P n

ðaþ1;bþ1Þ

¼ 12 ðn þ a þ b þ 1ÞP n1

ðtÞ

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321

we get D

mk

½ð1  tÞ

mþa

 1  X mþa ¼ ð1Þmþar Dmk ½tmþar : r r¼0

(6), (10) and again (11) establish ðtÞ P ða;bÞ m

m

¼ 2 ðt  1Þ

a

¼ 2m ðt  1Þa

(  1 X mþb k¼0 1  X k¼0

which yields the desired result.

k mþb k



mk



ðt þ 1Þ



mþa

X 1 

aþk

mk

r¼0

r

)  r aþkr ð1Þ t

mþa ðt þ 1Þmk ðt  1Þaþk ; mk

h

Many other well-known formulas for the Jacobi polynomials have analogies for the generalized case. For example, using (9) we get another useful representation of the g-Jacobi functions. We need the following auxiliary result [6]. Proposition 11. (Vandermonde convolution formula) For a 2 R, b 2 R, r 2 N holds the identity    r   X a b aþb ¼ : s rs r s¼0

ð12Þ

Proof. Considering the algebraic identity ð1 þ xÞ

aþb

a

b

¼ ð1 þ xÞ ð1 þ xÞ ;

expanding the terms in parentheses by the binomial theorem (11) and comparing the coefficients of the corresponding powers of x yields the desired result. h Theorem 12 (Explicit formula). The g-Jacobi functions can be represented as P ða;bÞ ðtÞ m

!

  1t ¼ 2 F 1 m; m þ a þ b þ 1; a þ 1; 2 m !   1 X m Cð1 þ m þ a þ b þ kÞ Cð1 þ a þ mÞ t  1 k 1 ¼ ; Cð1 þ mÞ k¼0 k Cð1 þ m þ a þ bÞ Cð1 þ a þ kÞ 2 mþa

where 2 F 1 ða; b; c; tÞ is the Gauss hypergeometric function (19). Proof. Writing t þ 1 as ðt  1Þ þ 2 and using (11), (9) becomes ( )    1 1  X X mþa mþb mk k r mkr m ða;bÞ P m ðtÞ ¼ 2 : ðt  1Þ ðt  1Þ 2 mk k r k¼0 r¼0 Rearranging the summands in the double sum yields     1 X s  X mþa mþb j mj sj ðt  1Þ 2mjsþj ðt  1Þ m  j s  j j s¼0 j¼0 ( s s    ) 1 X mþa mþb mj t1 X ¼ : 2 mj j sj s¼0 j¼0

ðtÞ ¼ 2m P ða;bÞ m

ð13Þ

ð14Þ

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Using (10) we arrive at ( ) s  1 s  X X m þ b t  1 Cð1 þ m þ aÞ 1 P mða;bÞ ðtÞ ¼ 2 Cð1 þ m  sÞ j¼0 Cð1 þ a þ jÞCð1 þ s  jÞ j s¼0 ( s  ) 1 s  X X mþb sþa t1 Cð1 þ m þ aÞ ¼ : 2 Cð1 þ m  sÞCð1 þ s þ aÞ j¼0 j sj s¼0 Applying (12) to evaluate the inner sum and using the definition (10) of the binomial coefficients with real arguments produces (14). To obtain (13) use (10), (5) and (19). h Theorem 13 (Differential equation). The g-Jacobi functions satisfy the linear homogeneous differential equation of the second order ð1  t2 Þy 00 þ ½b  a  ða þ b þ 2Þty 0 þ mðm þ a þ b þ 1Þy ¼ 0;

ð15Þ

d aþ1 bþ1 a b fð1  tÞ ð1 þ tÞ y 0 g þ mðm þ a þ b þ 1Þð1  tÞ ð1 þ tÞ y ¼ 0: dt

ð16Þ

or

Proof. Using the fact (cf. Section 5) that the Gauss hypergeometric function 2 F 1 ða; b; c; xÞ satisfies the Gauss hypergeometric differential equation (18) we get that 2 F 1 ðm; m þ a þ b þ 1; a þ 1; xÞ satisfies xð1  xÞ

d2 y dy þ ½a þ 1  ða þ b þ 2Þx þ mðm þ a þ b þ 1Þy ¼ 0: dx2 dx

ð17Þ

Substituting 1t for x together with (13) yields that P ða;bÞ ðtÞ satisfies (15). The proof is concluded by the fact that m 2 the left-hand side of (16) is a

b

ð1  tÞ ð1 þ tÞ fð1  t2 Þy 00 þ ½b  a  ða þ b þ 2Þy 0 þ mðm þ a þ b þ 1Þyg:



Theorems 10 and 12 together with some properties [13] of the Gauss hypergeometric function (19) imply further interesting properties of the g-Jacobi functions. Theorem 14. The g-Jacobi functions satisfy the following properties: (i) limm!n P ða;bÞ ðtÞ ¼ P nða;bÞ ðtÞ; m m ðtÞ; (ii) P mða;bÞ ðtÞ ¼ð1Þ Pðb;aÞ m mþa ða;bÞ (iii) P m ð1Þ ¼ ; m   mþb ða;bÞ (iv) P m ð1Þ ¼ ; m ðaþ1;bþ1Þ d ða;bÞ 1 ðtÞ. (v) dt P m ðtÞ ¼ 2 ðm þ a þ b þ 1ÞP m1 Proof (i) Using formula (13), it follows lim P ða;bÞ ðtÞ m!n m

   mþa 1t ¼ lim 2 F 1 m; m þ a þ b þ 1; a þ 1; m!n 2 m     nþa 1t ¼ ðtÞ: ¼ P ða;bÞ 2 F 1 n; n þ a þ b þ 1; a þ 1; n 2 n 

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(ii) Formula (9) yields ðtÞ ¼ ð1Þm 2m P ða;bÞ m

  1  X mþb mþa ðtÞ: ðt  1Þmk ðt þ 1Þk ¼ ð1Þm P ðb;aÞ m k m  k k¼0

(iii) Follows immediately from (9). (iv) Follows immediately from (9). (v) Follows immediately when we expand both sides according to (14).

h

To compare the classical Jacobi polynomials and the g-Jacobi functions a summary of their properties is provided in Table 1. If m is approaching to a natural number, the g-Jacobi functions become the classical Jacobi polynomials and their properties remain unchanged. Therefore, it may be concluded that the g-Jacobi functions extend the classical Jacobi polynomials. 5. F-Gauss functions Consider the Gauss hypergeometric differential equation xð1  xÞy 00 þ ½c  ða þ b þ 1Þxy 0  aby ¼ 0 which has as a solution the Gauss hypergeometric function 1 X aða þ 1Þ    ða þ k  1Þ bðb þ 1Þ    ðb þ k  1Þ k x 2 F 1 ða; b; c; xÞ ¼ 1:2 . . . k cðc þ 1Þ    ðc þ k  1Þ k¼0

ð18Þ

ð19Þ

(with the empty product assumed to be equal to 1), convergent for jxj < 1 [12, p. 63,14, p.283]. The idea is to generalize (19) by extending the differential equation (18) to a fractional order and finding a solution which could be a candidate for a generalization of the Gauss hypergeometric function 2 F 1 . Definition 15. We call the linear homogeneous fractional differential equation tm ð1  tm Þy ð2mÞ þ ½c  ða þ b þ 1Þtm y ðmÞ  aby ¼ 0;

ð20Þ

where y ðlÞ :¼ Dl ;

0 < l 6 1;

the fractional Gauss or the F-Gauss hypergeometric equation. Theorem 16. The series yðtÞ ¼ y 0 tq

1 Y k X gj ðqÞ km t ; f ðqÞ k¼0 j¼0 jþ1

0
ð21Þ

is a solution of Eq. (20), where fk ðqÞ :¼

Cð1 þ q þ kmÞ Cð1 þ q þ kmÞ þc ; Cð1 þ q þ ðk  2ÞmÞ Cð1 þ q þ ðk  1ÞmÞ

ð22Þ

gk ðqÞ :¼

Cð1 þ q þ kmÞ Cð1 þ q þ kmÞ þ ða þ b þ 1Þ þ ab; Cð1 þ q þ ðk  2ÞmÞ Cð1 þ q þ ðk  1ÞmÞ

ð23Þ

and q > 1 satisfies the equation Cð1 þ qÞ Cð1 þ qÞ þc ¼ 0: f0 ðqÞ ¼ Cð1 þ q  2mÞ Cð1 þ q  mÞ Proof. Let’s look for a solution of (20) in the form 1 1 X X yðtÞ :¼ tq y k tkm ¼ y k tqþkm ; q > 1: k¼0

k¼0

ð24Þ

ð25Þ

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Substituting (25) into Eq. (20) and using (6) yields Gy ¼ tm ð1  tm ÞD2m y þ ½c  ða þ b þ 1Þtm Dm y  aby 1 1 X X Cð1 þ q þ kmÞ Cð1 þ q þ kmÞ tqþðk1Þm  tqþkm ¼ yk yk Cð1 þ q þ ðk  2ÞmÞ Cð1 þ q þ ðk  2ÞmÞ k¼0 k¼0 1 1 1 X X X Cð1 þ q þ kmÞ Cð1 þ q þ kmÞ tqþkm  ða þ b þ 1Þ tqþkm  ab þc yk yk y k tqþkm : Cð1 þ q þ ðk  1ÞmÞ Cð1 þ q þ ðk  1ÞmÞ k¼0 k¼0 k¼0 Rearranging the terms in the sum, we get   1 X Cð1 þ q þ kmÞ Cð1 þ q þ kmÞ þc yk tqþðk1Þm Gy ¼ Cð1 þ q þ ðk  2ÞmÞ Cð1 þ q þ ðk  1ÞmÞ k¼0   1 X Cð1 þ q þ kmÞ Cð1 þ q þ kmÞ þ ða þ b þ 1Þ þ ab tqþðk1Þm  yk Cð1 þ q þ ðk  2ÞmÞ Cð1 þ q þ ðk  1ÞmÞ k¼0 ¼ y 0 f0 ðqÞtqm þ

1 X 

 y kþ1 fkþ1  y k gk tqþkm ;

k¼0

where fk and gk defined as in (22) and (23), respectively. Supposing that y 0 6¼ 0, in order to get y 0 f0 ðqÞ ¼ 0, q has to be chosen such that (24) holds. Thus, with y kþ1 ¼

k Y gj gk yk ¼ y0 fkþ1 f j¼0 jþ1

we see that GyðtÞ ¼ 0:



To justify the name fractional Gauss hypergeometric function, we show the relation between (21) and the classical hypergeometric series (19). Let m ¼ 1. (24) implies Cð1 þ qÞ Cð1 þ qÞ þc ¼ 0: Cðq  1Þ CðqÞ Using (5) we get qðq  1 þ cÞ ¼ 0 and, therefore, q¼0

or q ¼ 1  c:

Taking m ¼ 1, q ¼ 0 and y 0 ¼ 1 in (21), we get (19). Thus, we justified the following definition. Definition 17. We define the fractional Gauss or the F-Gauss hypergeometric function as the series 2

m

F 1 ða; b; c; tÞ ¼ yðtÞ;

ð26Þ

where yðtÞ is defined as in Definition 16 with y 0 ¼ 1. The following proposition was also established. Proposition 18. The Gauss hypergeometric function (19) is a particular case of the F-Gauss hypergeometric function (26). Acknowledgement This paper is partially supported by NSF – Bulgarian Ministry of Education and Science under Grant MM 1305/2003.

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