Kummer's twenty-four functions and n-fractional calculus

Theory,

N&incarA&ysis,

Methods

Pergamon

&Application& Vol. 30, NO. 2, pp. 1271-1282, 1997 Proc. 2nd World Congress of Nonlinear Analysts 8 1997 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0362-546X197 $17.00 + 0.00

PII: SO362-546X(96)00245-3

KUMMER’S

TWENTY-FOUR

FUNCTIONS

N-FRACTIONAL

CALCULUS

KATSUYUKI Institute Keywords

AND

NISHIMOTO

of Applied Mathematics, Descartes Press Co. Z-13-10 Kaguike, Koriyama, Japan %3

: Fractional Calculus, Ordinary Differential Equation, Gauss Equation, Gauss Hypergeometric Function, Kummer’s Twenty-Four Functions. Abstract

Many papers and books on the fractional calculus have been reported by the author already ( see the references). There are Kumuer’s twenty-four functions which are the solutions to the homogeneous Gauss equation. In this paper, it is shown that the solutions of Gaussequation obtainedbyour N-

factional

calculus

operator

Chapter

NYmethod

1.

cover

twenty-four

functions.

N-fractional calculus operator homogeneous Gauss equation

Q 0. introduction

(I ) Definition.

the Kummer’s

( Definition

(by K. Nishimoto)

of Fractional

N’method Calculus

to

)

( [ 6 ] Vol. 1)

Let ~=={4, D+), C-{C-,C+), C- be a curve along the cut joining two points C, be a curve along the cut joining two points D- be a domain surrounded by C- , 0, be a (Here D contains the points over the curve C ). Moreover, let f = f(z) be a regular function

z and - 00 + i In+), z and ~0 + i Im(z), domain surrounded by C, . in D (z ED)

,

(1) (m EZ’)

(f )-, - “@J(f)” where

-nsarg(f;-z)sn

forC_,

,

0 s arg( 5; - 2) 5 2a

(2) for C, ,

zEC, vER, r ; Gamma function, t *z, then (f ), is the fractional differintegration of arbitrary orderv (derivatives of order v for v > 0, and integrals of order -V for v -z 0 ), with respect to z , of the function f , if I(f)J < 00. (II)

On the fractional

Theorem

A.

calculus

operator

NY [ 2 1

Let fractiortul calculus operator (Nishitnoto’ s operutur ) N’ be

(v @Z-) [Referto (l)] 1271

(3)

1272

Second

with

World

N-”

Congress

-

of Nonlinear

lim NV Y--r-m

Analysts

and define the bijtary operatim 0 as N"dV"f -N’N”f -N’(N”f) rhcn

rl1c

9

(4)

(a,BER),

(51

(m =+I

set { N’}

=

(N”1v

(6)

ER}

is an Abeliart product group (having contirruous index V) which has the inverse tran sform operator (N’ )-’ - Nevto the fractional calculus operator N’ , for the furrclion -c m, v ER 1 , where f - f(z) atrd z EC. (viz. - 00< f suc/lfht fEF-[fjO*[f,I v coo). (For our convenience,

we call NP 0 N”

as product

of N’

and Nb .)

Theorem B. YIt;rre ” F. 0. G. (N ‘} ” is an ‘I Action product group which has co& calculus operator group > IIUOUSindex v “ fh- the set F . ( F.0.G. ; Fractional 51. N’ method to the homogeneous Gauss equation By our fractional calculus operator hry method we obtain the following sotutions which contain the N-fractional calculus. Theorem 1. Let Q,Ego = (q~ 10 */ rp,,I< W , Y a), theta fhe homogeneous Gauss equatior1 JqfP,z;u,B,Y

has solutions (GFoup I );

( Group

I

-~2~(z2-z)+qy{z(a of the form

(2 * fJ, 1)

-(z -l)y-/l-l)~-,

=ql),

9, - K(z-

-(z

= F(2),

(2)

QI = K((z

- lyl

T--r)a-l

=qg),

(31

fp = K((z

-1y-=-’

*2q-,

= qqS),

(4)

q7 = Kt’-’

(z-’

* (z - 1) -q,-

7 - PCs) ,

(5)

= p(6),

(6)

= cp(,),

(7)

- q,, 7

(8)

-lY-l)p~l

(denote)

(01

pl = K(F

(1)

ZZ ) ;

(p = fQ-‘(p

( Group

+JY+1)-y}+q7*ap-O

ZZZ ) ;

-@

-l>-")p-r

q.J = Kz’-‘((z

- 1)-P -z-y

fp = Kz’-‘((z

- l)-”

*z”-‘)8-r

0-v

Second

World

Congress

of Nonlinear

Analysts

1273

where tp,-d’q/dz’(k-0,1,2), Q+, ==cp-q~(z),zEC, constant, a, p and y m-e given constnnts. Proof of Group I; Operate N-fractional have then

calculus

operator

N*

and

directly

to the

K

both

is an arbitrary

sides

of ( 0 ), we

~‘~mJ;~,PJl~ =~2,,~(22-~)+~l+v~{z(2v+a+j3+1)-v-~} + aI, - (v2+v(a+p)+a~}-0 since

n N’(v,-9)=(F,*gy=c-,, I-(v

where Clroose

v

such

lhen

Substitute

(13)

OF” L-k (2” x

(14)

I-(v f 1) + 1 - k)T(k + 1)

(- 2’ U(O)).

II EZ,’

we have

(2 * 0, 1)

t

111.7

vz

+

Y(U

+p)+

u/Y-

arid

v=-u V = - cx into (13), yield

(15)

0,

-[I .

(16) (17)

fP2-,* *(z” - 2) -I-fp,-,, * (2 (-cl + /I + 1) +4x - y) = 0. ‘I’lierclcwc,

sclliq VI-,

==u -u(z)

(v

(18)

=%-I)

WC I1avc u +u *2(-u I from (17). ‘I‘he solution

of this

+/3 +l) 2

clilh-mmtkl

+rx - y

LqLtiun

u c KZ-‘+

-0

(19)

(2 i+o, 1)

is give

by

+-C-1.

(20)

Thusweobtain g’ - K(z”-’

. (2 - l)“-p-‘)a-l

(ckwole)

= q,)

(20) and (18). Where K is an arbitrary constmt. Inversely, the functiongiven by (20) satisfies(l9) clearly. (17). Therefore, the function ( 1 ) satisfies equation ( 0) .

(z*O,l)

(1)

from

For

V = - fl

the

, in

equation

the (0)

same

wny

(or

is different

from

Moreover, changing Vol.1 & [7]) from

the order za-’

( 1) when

the

change

of OT and

* (2 - 1)‘-”

other

ES V(2)

-qa-,

= K((z-

(a - l)@Z,’

and (t - l)‘-‘-I

in ( 1)

(

1 ), because

solution

(2 * 0,l)

l)~-~-y*Za-‘)~~, . In the same

we have

= fp[)) way

we have

v = K(( 2 - 1)“~=-’*Fr )#-, = qJ(s) from ( 2 ), which

,Cl in

equation

(2)

(I ), if u * p .

v

different

by

for U and /.,i ) we oblain

is symmetry 47 =K(P

which

merely

Hence ( 1) satisfies

is different

from

(2 ) when

(fl - 1) @ &+.

other

solution

(2 * 0 .I) other

([ 4 ]

(3)

solution

(2 * 0, 1)

(4)

1274

Second

World

Congress

of Nonlinear

Analysts

Proof of Group II; Set v ‘&, 4 - a) (2 * 0, 1) (Hence v, - AZ”-‘@ + z”& and v2 = A(A - l)zA-2@ + 2At”-‘& + zAti2 ). Substitute (21) into (O), we have then

(21)

~~.(Z2-Z)+91.{Z(IX+Bt1+2A)-2~-Y)

+g,(~(~-l)+A(*t~+I)tu~-1-‘n(n-lty)j-O where $ - d’glk’ (k = O,l, Here, we choose A such that

2)

and

q5,, - 0.

A(A -l+ that is,

(22)

y) = 0

A-0,

(23) (24)

l-y.

(i)

Inthecnse A -0 this case we have the same results as Group 1 . (ii) Id the case A = 1 - y Substituting A - 1 - y into (22) we have In

~2~(~2-~)+$l~{z(a+/3-2y+3)+y-2}

+#.{(l-y)+a){(l-y)+/3)-0.

(25)

Next, operate N” to the both sides of (25), we have then @2+,

*tz*

-

{z(a+(J-2y

e+9,+“*

+3t2v)+y

-2-v}

+~“+~+V(U+~-2y+2)+(1-y)(a+p+l-y)+u/3}-O. Here we choose

v

(26)

such that

v2tV(Ut/j-2y+2)+(1-y)(U+~+l-y)+Up=u,

(27)

that is,

v-y-a-l

(28),

v-y-p-1

and

(29).

1) For the case of (28); Substituting (28) inlo (26), we have

~,+y-o~(22 -z)++~-~

.{z(p-

a+l)+a

-1)

(30)

-0.

Set &-cl -u-u(z)

(31)

p=u,-,>,

we have then

u;(z”-z)+u-(z(P--++)+a from (30). Thesolution

to this differential

u - Kz”-‘(z where K is an arbitrary Therefore, we obtain

v - Kz’-’ (34)

and

- 1)-8

by

(33)

(z*O,l)

constant.

# - K(P*(z from (33) and (31), hence we have from

(32)

-l}=O equation is given

-1)-q-r

(za-1 -(z - 1)-q-”

(z*O

- q,,

(34)

31) (z*Ootl)

(5)

(21).

Inversely, (33) satisfies ( 0), since we have (21);

(32),

then

(34)

satisfies

(30) clearly.

Therefore,

(5) satisfies

2) For the case of (29); In the same way as 1) ( or merely by the change a and p in (5 > ) we obtain

Q= w

(ZB- *(z - I)-“)#-,

- Q)

(2’

0 91)

(6)

Second World Congress of Nonlinear Analysts as the solutions Moreover,

to the equation

cbnnging

(0),

the order

which

zn-’

is different

nud (t -l)-’

’In (5)

1275

from (5)

if a * p.

we have

other

solution (7)

different

from (5)

when

(a - y)@Z,’ qJ = Kz’-’

from (6),

which

Proof of Group

is different

. In the same way we have ((z -1)~”

from

(6)

.zJ-l)b

whm

(fi

-r = q$)

-y)eZ,’

other (ffO,l)

.

#-Qi(z)

(35)

(z*(J,l)

lhcn

+~+-1)-‘(A~+AU+A~-Ay)+(A+a)(13.+&0.

(36)

we choose A such that A(A+a+f?-y)-0,

that

In the caseA In

tllis

CCISL’

WC

y -a-/J.

(38)

-0 the

have

(ii) InthecaseA-y-u-p In this case, substituting Next

(37)

is, A -0,

(i)

RBIIIC’

rcsdls

Group

ns

1.

A - Y - a - p into (36) we have

-a-p+l)-y)+#-(y A e* - 2) + $4 &(2Y operate NY to’ the both sides of (391, we alive 422c” *(z’ -z)+

Here,

we choose

v

4,”

*(z(2v+2y

+$V+z+*(2Y SUB that

-a

-a)(y

(39)

-P+l)-v-y}

-a-P)+(y

-a)(y

-B)}-0.

(40) (41)

is, v-a-y

v-p-y.

and

(421,

1) For the case of (42) ; Substituting (42) into (40), we

(43)

have

(44)

Rcamr -(z” - ~)+~l+.-r-{z(~-~+l)-at-O. Next

-p)-0

then

vZ+v(2y-a-P)+(y-u)(y-/3)-O, that

(8)

III ;

Set v -(z-l)A!+, aud substitute (35) into (0). we have $t%~*(1~-t)+9;{Z(U+P+1+2A)-Y) Here,

solution

set #*+o..l - w - ‘V(Z)

we have

(@-w

7 -a-1

1,

(45)

then w,~(z”-t)+W~{z(Q-P+l)-Q)-O to this ectuntion is giveu by w - Kz-“(z -l)“-’ arbitrary constant.

(46)

from (44). Abe solution where K is an Hence we have from

(47) nod (45). Therefore

4 -K(z-Yz we have

-d-I),-,-,

(47)

(48)

1276

Second World

(p -K(r

of Nonlinear

-1) ~-12-qZ--a -(t-l)fl-‘)r-a-*

from (48) and (35) which Inversely, (47) satisfies ( 0), since we have (35).

v -K(z changing

Analysts

- qg)

has A = y - a - /3, (46), then (48) satisfies

2) For the case of (43); In the same way ( or merely

Moreover,

Congress

(denote)

(44) clearly.

-l)q&-,

-((z

-l)“-‘z-“)r-n-,

(9)

(9) satisfies

) we obtain (2 # 0,l)

“fq,a,.

the order Z-” and (z - 1) b-1 in (9)

ql =K(z-1)

Therefore,

by the change a and /3 in (9)

-l,f-a-a(z-p(z

(2 # 0,l)

(10)

other solution

we have

E qt,r,

(2 * 0% 1)

(11)

“q*2)

(2 so ,l)

(12)

which is different from (9) when (y -a - 1) $ q. In the same way we have other solution v =K(z from (lo),

which

Theorem2

-1) “--Tc,

is different

from (10) when (y - fl - 1) $! 20’ .

a - /3 , we have the fillowing

Wlze~r

%

a P(2)

P(7)

= qs,

’ ’

respectively. Proof. It is clear, because Theorem

-rq+,

3. We have

and respectively.

the @lowing

and Proof.

%

-

9

-q(6)

%)

and

p7(10) )

Qw

each other when

* P(l2)

(49)

’

(50)

f

a - /J , respectively.

iderttities,

(a -1)EZi

,

(51)

(P--w=,'

'

(52)

%I - %)

fir

Q)(Z)=

47(4,

f

%)

=

V(7)

Jbr

(a-W%,

(53)

%)

= %I

f OY

v- YF&

(54)

%, - qll)

f or

0

(55)

%J)

f Of-

(Y-B-v=,"

“5912)

and

v

Ql(l)

-

P(2)

-

(p,3,

-

V(4)

p(5)

-

4)(6)

-

97(7)

=

v(S) -

--We,

VEZ,’

(56)

then ,

(57)

for UfV, and v~Z;. clearly. ( [ 7 J Vol. 1 & [ 8 ] )

following

%I - Q)(lO) - qll) It is clear by Theorems

-a

we have

for

(U’V>, *(v*u>, we have this theorem 4. We have the

OT

=v(z)E@‘,

(u ‘VI, “(V’U),

Theorem

= 9)(4)

they overlap

Proof. Let u =U @)E@” and Therefore,

V(3)

identities,

P(12)

identities, for

a-P,

fur

a=

P,

for

a-

/3, (y-

2. and 3.

(58)

(a-l)E$,

(59)

(a-

(60)

YFG,

a-l)EZ,’

.

(61)

Second World

Congress

of Nonlinear

1277

Analysts

Chapter

2. More familiar forms of the solutions obtained in Chapter 1 and Kummer’s twenty-four functions 5 1. More familiar forms of the solutions Group I in Chap. 1, 5 1. ‘I’heorem 1. By the fractional calculus of products (using the generalized Leibniz ncZe)wehave

([6lVol.l

q.

*(z - l)-)a-l

= K(P

& [7]> = 2-I (l-z)+‘,F,(l+p-y,l-a;2-y;-3

(1)

- %J) for l(za-7)~-1-~I~~ (nEZ’U{O}),z*O,l and Iz/(z-l)l
(2) _Ke-i”(u-l),l-r(Z-1)‘-8-i~

r(a)r(y n-on!T(a-n)r(y

-1-n)T(B+ 1- y +n)(l)” -a)r(p+l-y) 2-l

(3)

aKe’“(’-0-8)r’y-1)z1-r(1-z)r-b-1*1~(pfl-y,1-a;2-y;~) w -4 .

under the conditions,

since

r(a)r(l -a) r(,r+l-a)

r(a-fr>-(-l>-” Therefore,

(4)

I

(nEZ’U{O}).

(5)

choosing

K=lIM

( M-e

in(~-9-yy-iyr(y - a))

(6)

wehave(l)from(4). By the change of a and /3 in (1 ), we have z l-r(l-t)‘-“-‘,F,( Theorem

(7)

2. By the fractional

q3) = K((z -1)“~‘-l

.2.-1),-l

calculus of products, we have = za-’ (1-z)y-“-8,~(l-a,

y-a;l-a-@+y

;l-i) (8)

= vu, forl((z-l)r-~-l)=_l_"lcm Proof.

(nEZ+U{O}),

z*O,l

and[(.z-l)/z)
We have

(2 * 0 71) I Ke~=(l-8-1)ZU-l

(9)

(1 +P.-B~ Il.0

n! r(a

=Ke in(r-8-l) r(a +B - Y) za-r (1 -Z)-8 w + B - Y)

-

n)r(i +fi - y)r(y -a)

zF, 1-a,y

-a;l-a-/3+y;l-i)

z

(11)

1278

Second World

under the conditions.

Therefore,

Congress

of Nonlinear

Analysts

choosing

K-IIM ( M-e ‘n(“-fl-l)r(a we have (8) from (11). By the change of a and /3 in ( 8 ), we have $-‘(1-2)‘-a-f12<

(

+/%y)lr(l+py)),

(12)

l-p,y-p;l-a-B+y;l-+=~~,.

(13)

3. Without the me of generalized Leibniz rule, we h&e

Theorem

qlj = K(r’-’

.(z -l)r-8-l)~-,

1

(Y -a,/I;p-a+I;-

= (1424

(14)

1-Z

= q5),

for)((1-2)"-8-'-').-11
(kEZ’U{O}),

and jl-~l>l.

z*O,l

Using the identity

zA=(z-l)A(l-~)a

*(z-l)A.&($(;=:’

k)(l-Z)-k

(ll-zp1,

(15)

we have ~(1)-K(za.(z-1)y-8-1)0_1

=Kz

@*a-y)

(-l)V(A

+l)

k.o ryk + 1y-p

+ i - k)

((l-

z)-‘(z

(-i>“r(a

(z*O,l) - 1)““~fl-‘)

-y +l)r(p

(16) (17)

O-l

+k)

(1%

(20) under the conditions. Therefore choosing

K-l/M ( M -ei”@-8-“I’(/3)lI’(#3we have (14) from (20). By the change .a and /3 in (14), we have

a +l))

(21)

(22) Theorem

4. Wthout the use of generalized Leibniz rule, we have

~(1) - K(z”-’

~(z-l)r~P~)~~~~(1-z)“~a~A~~(y-a,y-~;1-a-~+y;l-z)~23~ -
for 1((1-~)~+~+~)J~ Proof.

00 (kEZ+U

{0}),

z*O,larzd~l-zl
Using the identity t* - (1-(1-z))”

we have

-

r(;;y;z;;;tk)(l-z)k

(jl-zl<

1)

(24)

Second World Congress of Nonlinear Analysts

1279

(25) (26)

k-0

- Ke’“(’

r(k

-

4-l)

+ l)r@

(27)

+ 1 -k)

wkr@

- Y + wa

+p - Y +)

(1

+k+r-ii-B

cm

c -or(k+i)r(a-y+i-k)r(l+B-y-k)

-a,y

-j?;l-a-fl;l-z)

(29)

under the conditions. Therefore, choosing Ks.lJM M ~e’~(~-~--l W+B-r)iw+B-Y)) ( we have (23) from (29). By the change a and /3 in (23) we have (23) itself again. Theorem

5. Wirhout the use of generalized Leibniz rule, we have

fp@)= I+-

* (2 - l)+l)a-l

fO+“-8-1-*)~e11
(30)

=(--t)-‘&(B-y

+l,#I;/I-a+l;l)

2

(31)

(kcz*U(O~~~:o,lond,~(rl.

Using the identity

(32) we have

~(1)= K(z’-*tz-l)A)a-l (-i)kr(Y k-0

w

-

+ i)w

(A==y i?

+

-B--l)

(ZfO,l)

(33)

(za-B-1-k) -

k)

(34)

Cl-1

m 1 -Ke-iaaZ-fi

(35)

c k-~r(~+i)r(y-B-k)r(B+i+k-a)Z-k

I -Keis(B-a) under the conditions.

(36) Therfore, KS l/M

choosing ( M w -..$+-=

QB)/r@

-a

+I))

(37)

-J$,.

(38)

we have (31) from (36). By the change of a and /3 in (31), we have (-t)-”

Theorem

6.

&

(a -y+l,a;a-B+l;i)

z

Without the use of generalized Leipniz rule, we have

~~1)-K(~~~7~(~-l)y-B-l)o-l=~1-~2~(a-y+l,~-y+l;2-y;z)

(39)

1280

for

Second World

I(z~+~-~)~-~-

Proof.

Congress

(kEZ+U(O}),

of Nonlinear

z*O,l

Analysts

and li+l.

Using the identity (40)

we have

(d-y

-/3-l)

(z*O,l)

(-l)“r(n + 1) (Z&+o-r) o J&WY -6-l) c T(& + 1)Iy + 1 -k) a-l k-O OD (-l)‘r(Y -w(Y -1-k) in(y-a-8)Zl-y -Ke Ic

(41) (421

k

(43)

k-‘, r(k + l)r(y - p - k)JY(y - a -k)’

-Ke

in@-a-fl)

UY - 1) *I-Y w

under the conditions.

&(a-y+l,p-Y

+1;2-Y

;Z)

w

-a)

Therefore,

choosing

K-IIM,

i”(y-=-B)r(y --I)/r(y - a))

(M-e

we have (39) from (44). By the change a and p in (39), we have

(39) itself

(45)

again.

5 2. Commentaries (I 1 When none of the numbers Y , a - fl, Y - a - /Y, is equal to an integer, each of the following twenty-four functions (due to Kummer ) satisfies the homogeneous Gauss equation 5 1, (0) in Chap. 1. [ 81. List of the twenty-four Vt1,=2Fh,

P; 7; 4

vOl=(l

-4Y-“-b2Fl(y-a.

v,,,=(l

-z)-“J,

Vcsl=2Fl(a, y-0;

-4-b2Fl

y; 2)

v,71=z-‘2F,

z-l

y-a,

~~8,=zeb2Fl

I-Z) l--JZ >

P+I--y,B;a+/l+l-y;

>

a,a+

1-z)

(

j?; y; 5

~~9,z(-Z)-“~Fl

I --y;

a,a+I--y;a+I]+l-yy;

>

(

I---!-

(

a+

I-y;

I --/I;

Z >

i

( ~~IoJ=(-Z)~-‘(I

P; a+/?+

Y,~~=z’-Y~~,(a+I--y,~t-I-y;a+~+l-y;

y; LT.-

a, y-p; (

yOj=(1

functions by Kummer

Z > -Z)‘-“-ptF,

I -/I,

y-/l;

a+

I -a;

(

+ >

1 V,~,j=(l-z)-“J,

a, y-j?;

a+

I -/?;

V~,~)=(-Z)‘-r(I-Z)Y-=-’

__

l-z

(

2Fl

a+l-y,

> 1-P;a+l-/I;-$--

(

-z

j+l--y,LJ;/J+l-a;I Z >

>

Second World

Congress

V (‘4)=(-Z)ar-~(l--Z)“-~-~ZF~

V,,,,=(I-r)-“,F,

(

(

V (zob--z’-~(I Vo,,=(l

we have

>

l-fl;2-y;z) (

a+ I-y,

1 --/I; 2-y; *

/?+ 1 --y,

I

( -z)7-u-P2FI(y-aa,

y-p;

V~22~=z’-Y(l-z)1-‘-P2FI(I-a,

Moreover,

I--rr;p+l-a;&

I -y, p+ I -y; 2-y; 2)

-z)~-“-~~F,

( l-z)‘-

V ,zLj=~P-Y(I

>

>

/?+I-y,

-z)~-~-~~F,

V (*j)=z=-y

1281

Z

V llw- -z’-y(l-z)Y-‘-CZF1(l-a, V l19)- -~‘-~(l

Analysts

I--or,?--a;p+,-ol;J-

/~,y-a;(l+l-a;-~~l-

V(16)=(-z)‘-r(l-Z)Y-P-l*F, V t171- -z’-YzFl(a+

of Nonlinear

-z)~-‘-~#,

the following

-a; 2-y; z-l

y+ 1 -a--P;

(

y-p,

>

I -z)

I-p;y+l-a-/?;

m-c*F, y-a, (

) Z

l-z)

I-a;y+I-a-j;

l--!Z

1 -p; y+ I -a--p;

I -+-

> >

.

six identities.

( i ) 5,) = y2) = q3) = y4) f

(iv)

I&,

(ii ) J$, = F6) -q7)=qsjr

(v)

v&)=yl*)

= Y19) = qIl) 9

(iii)

(vi)

&=~22,

-qa)

q9, - G;lo, = q11) - y12, ’

- q14) - &;ls) - Tl6) 9 ==~zA)-

(II) By our N-fractional calculus operator N’ method to the homogeneous Gauss equation we obtained the solutions shown in Chap. 1, which have the fractional differintegrated forms respectively. The translations from our solutions Chap.1, Group I to the more familiar forms which contain the well known Gauss Hypergeometric functions yield, as we see in Chap. 2, § 1,

3 V(U), c&), yn) $0,~ J&,1 q23,~ VW’ T/;ap V(11) ’ V(21) (refer to the list described above ). In the same way as the procedure shown in Chapter 2. Q 1, the transrations solutions of Group II yield and the transrations

y3,p y4,p yz4,p %V, &), V(16)’f&z), y9,, from the solutions of Group III yield

f&9

from the

y*,,

vm,, qzo,, q7,y yap ynp J&)7 J(5), J$3), y14p J(M) * Therefore, we see that almost all functions of the group I$, -+ I&, in the list described above can be derived directly from our solutions of the Groups I, II and III in Chap. 1, which have fractional differintegrated forms, except only two functions I&, and y6, . (For the calculations from the solutions 10, November 1996, pp 9 - 23. )

of Group II and III, refer to J FC Vol.

1282

Second World

However,

we have

Therefore, in

Chap. That

the

relationships

we can derive

of Nonlinear

Analysts

( i ) and ( ii ) respectively. twenty-four functions

the Kummer’s

from

our

solutions

1. is, the

solutions

obtained

functions. (III) All mathematicians and

Con8ress

N-fractional

calculus

equations with that Weyl, Osler, Oldham

by

our

should

compare

operator

Nvmethod

NY

of other fractional and Spanier’s ones.

operator

method

cover

the

Kummer’s

24

our N-fractional calculus, N-transformation to the ordinary and partial differential calculus, for example, Riemann-Liouville,

(IV) Hitherto, only the solutions of the formsin Chap. 1, Group I had been treated in the applications of fractional calculus to differential equations. However, this is insufficient. Namely, we must add the solutions such as the forms of Group II and III, to the other differential which are shown in Chap. 1, Q 1. For the solutions equations the situations are same.

References 111 K.

Nishimoto; (1993), 29-37.

Solutions

I21 K. Nishimoto; Calc.

of Gauss

On Nishimoto’s Vol. 4, Nov. (1993). 1-11.

equation fractional

in fractional calculus

calculus,

operator

J. Frac.

Calc.

KY ( On an action

Vol.

group)

3, May J. Frac.

K. Nishimoto; On the fractional calculus of functions (a -z)’ and lOg(U -z), J. Frac. Calc. Vol. 3, May (1993),19-27. function, Solutions of homo eneous Gauss e uations, which have a logarithmic 141 K. Nishtmoto; in fractional calculus, J. Frac. Ca Pc. Vol.5, May (I 994),11-25. Some properties of N-transformation, J. Frac. Calc. Vol. 8, Nov. (1995),1-10. 151 K Nishimoto; I31

161 I71

K Nishimoto; Fractional pf~;~~~~~$gvg~&p.

Integrals and Differentiation 181 W. Magnus, F. Oberhettinger

Calculus,

Vol. 1 (1984),

Vol. 2 (1987),

Vol. 3 (1989),

Vol. 4 (1991),

tshmoto’s Fractional Calculus ( Calculus of the 21st Century); of Arbitrary Order (1991), Descartes Press, Koriyama, Japan. and R. P. Soni; Formulas and Theorems for the Special Functions

111 B. Ross; Methods of Summation(1987), Descartes Press, Koriyama, ; Fractional Integrals t 121 S. G. Samko, A. A. Kilbas and 0. I. Marichev of Their Applications (1987), Nauka, USSR. [13] K. S. Miller and B. Ross; An introduction to the fractional calculus equations (1993), John Wiley 61 Sons. Inc. A plications, [14] V. Kiryakova; Generalized Fractional Calculus and F’itman-Longman( co-publ. John Wiley & Sons, New Yor I: ) (1993).

Japan. and Derivatives, and fractional Research

Notes,

and Some differential Vol. 301,