V.A. DITKIN and A. P. PRUDN~KOV

(MOSCOW)

(Received 25 tfay f962)

Introduction

The methods of operational calculus are very effective in the solution of many problems in applied mathematical analysis. It is usually the differentiation operator p = ri/t/dt which is considered. In this article we shall consider operators of the form

uh. = .+ ( t $ )“, k = 2,:s ,._., R,

which are closely associated with the Ressel equation or its general- ization to the case of higher order equations. Therefore operators of this form can conveniently be called Ressel operators cl]. Suppose the linear set M is formed by functions f(t) defined in the region n with the ordinary operations of addition and multiplication on real or com- PI ex numbers.

As we know, the 1 inear set df is called a ring if for any of its ele- ments f, g associative multiplication f * 6’ E !l, which is permutational with multiplication by numbers and distributive with respect to addition, is defined. We shall consider only commutative rings with a unit and without a divisor of zero. Any such ring h! can be extended to a field S? (the elements of the field are denoted by f/b’, g f 0). ‘Se have I‘/& = fl/gI when, and only when f * g1 = f, * g. Following Mikusinskii [21, the elements of the field m will be called operators. Let numbers belong to the ring hf. In this case we can consider the product of a number h by the function f f !! either as the ordinary product hf, or as the product of elements of the ring h * f.

We shall only consider rings in which both these products are the

l Zh. vych. mat. 2, No. 6. 997-1018, 1962.

1181

same, i.e. A * f = hf. In this case h = 1 is the unit of the ring ilf. The operators of the form h/p (h and u are numbers) coincide with ordinary fractions h/p. The operators of the form j/l coincide yrith the functions of the ring II. In the field %? we can consider operators de- pending on a parameter. Such operators are called operator functions. The basic concepts of mathematical analysis, the concepts of a limit, continuous function, derivative and integral can be carried over into operator functions. To do this it is necessary for these concepts to be introduced in the initial ring bf. We shall therefore assume that the ring irf satisfies, in addition to the preceding conditions, the following additional requirements.

1. The region f, is the ha1 f-line 0 -< t c a.

2. Each function of the ring ?I is continuous in the region n.

3. If the sequences of functions j,(t) E IV and g,(t) E M respect- ively converge in the region 9 to the functions j(t) E M and g(t) E df and in any finite interval 0 < t < T the convergence is uniform, then

4. If f(t; A) depends on the parameter a < h ( F and for all a < h < p, /(t; A) f !f, af(t; h)/;)h E h! exists and the function ?j/?h is continuous with respect to the variables t and A in the region o<t<@J, a < h ( p, then for any function g(t) E hf we have

5. If f(t; A) is continuous in the region 0 x< t < m, aI < h ( F1

then for any function g(t) E nf

F :j

\[/(t;h)“‘(t)]dh =(f(l;l)dh*g(t) (!q’<x<p<pl). rr

J a

If f.2 is any operator of the field Ills, an operator 9 such that the product 1 * a If now n depends on the parameter A, then ator 7 will also depend on A.

then we can always pick out belongs to the initial ring M. in the general case the oper-

f?efinition. The operator function a(h), a < h < P is said to be ducible on the interval (a, j3). if there exists an operator 7 which does not depend on h such that for all h. a < h < F the product

q*a (A) = cp (t, A)

belongs to the ring 41.

re-

1186 V.A. Ditkin and A.P. I’rurlnikov

We can always assume that the operator q belongs to i.e. is a function belonging to the set M. The sum and tions which are reducible in the given interval (u, p)

the initial ring, product of func- are again re-

ducible functions. Thus, reducible operator functions form a sub-ring in the field of all operator functions. It is easy to extend the basic operations and concepts of mathematical analysis to reducible operator functions [21. This is done according to one general rule. A reducible operator function is said to be continuous if there exists an operator 7 (7 does not depend on A) such that q(t; A) is a continuous function of two variables in the region 0 < t < 0~; a < h < p.

A reducible operator function n(A) is said to be differentiable in the interval (a, p) if the function q~( t; A) is differentiable with re- spect to the parameter A. If, in addition, a@?, is a continuous func- tion with respect to the variables t and A, then the operator function n(A) is said to be continuously differentiable. If n(A) is a continu- ously differentiable operator function. then the operator (l/7) (%-p/ah), where q~ = 7 * a(A), is said to be the derivative of the operator function and is denoted by

4 acp a’ (A) Czz 9 = - -* q 0)”

This definition does not depend on the choice of the operator 7. The derivative of an operator function possesses the properties of an ordinary derivative; namely, if the operator functions a(A) and b(A) have in the interval a < A < p continuous derivatives, then their sum and product possess continuous derivatives in this interval, where

Ic (A) + 6 @)I’ = a’ (A) + 6’ (A)

[a (k)*b (k)]’ = a’ (A) * b(k) + a (5) + 6’ (A),

Q(A) * [ 1 bcn)= a'(A)*b(~)--o(A).b'(h) b(M*b(U

assuming the existence of the left-hand side of the last equality. For reducible operator functions n(A) the integral is defined

P

\ a (A) dh.

. a

If n(h) is a reducible operator function on the interval a1 < h C ;1 and a1 ( a < ,‘, < p1 then, by definition, there exists an operator 7 E hl

such that 7 * n(A) = cp(t; A) E by, a1 < h < F1. Uy definition we have

1187

The integral of an operator function possesses the properties of an ordinary integral.

The sequence of operators an is said to converge to the operator a if there exists an operator 7 such that

QWZ, = fn (t) E M, g*a = f (t) E M,

and the sequence f,(t) converges to f(t) uniformly in any interval o<t\<T.

In this case we write

No convergent sequence of operators can have more than one limit.

If

Iim an = Q, limb, = 6, lZ-+CO n-w

then tfre sequences an + b,, a,, *&, will be convergent and

lim (a, -+ I,,) = a + 6, lJ-.M

lim a,*b, = a*b. n-rcx,

If the limit lim >, n-em 11

exists then this limit is equal to +.

In the next sections we consider certain functional rings which are special cases of the ring b! we have been discussing, and we study the corresponding operator fields. By analogy with the calculus of the differentiation operator p = cl/& we could develop the theory of Bessel operators much more widely t.AM we do here, particularly with regard to operations with operator functions and operator series and the solution of differential equations. The notation for the special functions and certain constants we shall use was given in [31.

1" Ihe functionai ring 31, w

bet Ffnj be the set of all complex functions f(nl, n2, . . ., xn) of

n real variahl es defined in the region /?(“’ (0 < x1 < ED, . . . , 0 < xn < 0~)

and Lebesgue integrable in any finite n-dimensional parallelepipid

/?j:,‘. ,,..,. “,, (()<.rl <<I, (1 <.r2<flz ,... , (lL<.Tn.<fl,). In the set F(,)

let us pick out the sub-set L, of functions of the form f(rl x2.. ~“1. We know that the integral

XI X2 . 11 (z-1, 52, . . . > 22,) =

c\ . . .

i, b . . . , xn - En) dE,& . . . &. (1.1)

is called the n-dimensional bundle of the functions f(x,, x2, . . . , xn)

ad g(x,, x2, . . ., xn).

Let f(x,, x2, . . ., xn) = f(x, x2.. .xn) and g(nl, x2, ,. ., xn) = &(X1 x2.. .X”). Then their n-dimensional bundle takes the form

In the last integral let us make a substitution of variables accord- ing to the formula ik = x,u,(k = 1, 2, . .., n). Then we find

or

If we nut

h(t)= \dU,i du,. . . ~f(u’U’...Un~gr(~-u~)(*-un,)... il

. . “(1 - un)] dun, (i.2)

then

%a

fP s . . . ff%rEz...E*)gI(Zf-El)(sz - E2). .* f% - EtJl4l4a..*4‘sL=

Oi a

= h (zpz2 * . * 2,). (1.3)

Thus, the Set of all functions of the form f(xln2.. .nn) forms a ring

with respect to the operations of addition and contraction. To each function f(x1x2.. . xn) of this ring there corresponds a function f(t) of one variable t. The set of all such functions is denoted by f., . If the

function J(t) belongs to the set LB then, by definition of th;s set, n

4 1

ii s . ..an.j(z~~2...z,),dj,~~2...cE~,<oo.

iG 0

Let us make certain tr~sformations of the last integral. After sub- stitution of the variable using the formula zn = anuR we obtain

The subsequent

.I J J 0 0 0

substitution x”_~ = a, lzz, I gives _ _

00 0

Qn-2

Z.Z 5 dx,-, \ a,, dun-, 0 fi

Continuing in this way, successively introducing new variables according to the formulae

1191) t’. ,t . ?itkin and A.?. t’t.orfnikov

we obtain

T’s s . . . *~~1(512%...~~),~~ldz,...dz,=

no 0

a,a,. . .un

= s du&,\du,. . . ~,1(uIU2...un),du,.

0 0 0

Introducing the new variable r = uiu2.. . u,, in this integral we have

Hence, the set L, consists of all the functions f(t) defined on the

ha1 f-line 0 < t < 0 2nd satisfying condition (1.4). It follows from (1.1) that

Applying this inequality to functions of the set L, aud using (1.3).

(Lb) we find n

a,~*. ..a,

x s !.#$!* _ .~~~ua~‘~-;g(Ua),dun. 0 0 0 0 0

me bundle defined for fttnctions of the set L, according to formula

(1.2) Possesses aI the properties of an ordina~~b~dle, viz. :

Opernt ionn 1 calculus of Resse 1 operators s191

1) conmutativity, i.e.

t 1 1

'j s dul l du,. . . \f(U~z+.. u*)g[(t - 241) (l-u*). . .(I- un)]durc =

a 0 1

t 1

=

s I dUIL,. . . \g(u’z+.. 4 f [(t - ~1) (1 - u,) . . . (1 - un)l du,;

0 0 0

2) associativity, i.e.

3) distributivity, i. e.

f 1 .

I 5 dil’dT&. . . \ f (E&2 * * * En) (47 IQ - Ed (1 - s4 * * . (1 - En)1 +

l l 0

+ h r(t - Ed (I- Ea) . . . (I- En)11 dEn =

=\ti$W.. . ~f(I~E....S$gI(~-~~)(~-~E,)...(l-h)l%+

r(t - Ed (1 - Ea) . . . (I- Ml 4.;

ilar properties of an n-dimensional

a 0 0

(These Properties follow from the sim bundle of functions of n-variahles. )

4) if for all t >O

1192 v. .1 . Pitkin and A.r. Prodnikov

t

s s d%, ’ d%, . * En) g re - El) (I - Es) * * . (1 -Ml d%,, = 0, (1.5) 43 0 0

then at least one of the functions f(t) and g(t) is equal to zero for almost all t on the ha1 f-line 0 < t < a~.

The last property follows from the theorem of Mikusinski and kyl’ - Nardzewski [41. In order to use this theorem it is sufficient to see that (1.5) is equivalent (see (1.3)) to

.x1 x2

1s . ..s” I (%I %z * * . %n) g [(XI - Ed (2, - Ez) . . . (xn - %n)l d%, 42 . . . d%n = 0 00 0

for all O<zi<m (i=1,2,...,n).

From (1.1) we have

x b” (XI- %I, xz - fz, . . . , zn - %,a) d&t =

f (El? Ez, - * . , En) d%, dE2 . - . d%n x

n, a?

x \ \ . . . s” g (x1 - El, x2 - &, . . . . x,, - &,) dxl dxs . , . dx,,. ;I L E,

Here we have made a change in the order of integration using the formula

Operat ionaf calculus of‘ ijesse 1 operators 1193

an

dx2 . . . s m (x1 x2 . . . x‘, ilkc . . . En) dx,. 47,

Hence,

where

G(x,, 21, . . . . xn)=r.~...ip(ll,5,,... &)d$&z...dEn. 0 0 0

Applying the latter equality to the functions of the set noting that

we find that if

then

. .

l”Bn and

1194 V..t. Ditkin and A.P. Prortnikov

where

Let Ile denote the sub-set of all functions F(t) which can be put in

the form n

where f(t) E IA, and C is any const‘ant. n

The function f( t) belongs to the set L, . F’or every function of the

set YB n th there exists almost everywhere the n derivative F’“‘(t). In

the set” !?n let us define the operation of mu1 tip1 ication by the formula

P (1)*G (t) 2

If

V,“?. . .un__l

\ f (un) dun ,

I, (I II 0

We define addition in t!:.e set ;!,,I in the nat.urd way. The product. tle-

fined in - ( 1. fi) again be1 ongs to ;lhe set !‘,, , it is commutative,

associative and possesses distrihtrtivity with respect to sdditim. Tliere

fore t!le set f’ R forms a commutntive rin(:. Here, if one of t!le factors

in (1.6) is ~~~enn,~~~,e~ cx, then we tinve

“perat ional co lcu lus of Resse 1 operntors 1195

=u -&j ’ d\d$[‘F(U,)du,=aB.;F(t)=aF(t), 2, 6

where B = -$t Gl (see [51), i. e. the product of a number by a function

in a ring coincides with the ordinary product of a number by a function.

If both the factors in (1.6) are numbers, then the product (1.6) is the same as the ordinary product of numbers. The ring !YB has no divisor

n of zero.

We can extend

extended ring by

F’or the operator inverse operator

2. The operator

the ring I!!~ to the n

Bn=+(t$)n

field of ratios. Let us denote the

all,. We call the elements of the set al:,, operators.

l/t we introduce the notation l/t = Sn. Then for the 17;’ = l/B, we shall have l/B, = t. Therefore

$ j(t) = f(t -qjd%l!j de,\ d& . . . i &Ez.. . n 6 ; 0 0

* - * %nf rtt - %l) (1 - Ed * * * (1 - %n)l d%n,

or

f (Un) dun (2-l)

If F(t) E MB and F(O) = 0, then ”

BnF(t)= +(I $)“F(t). (2.4

It follows from (2.1) that

lJ!lr, V.A. Ditkin and A.P. Procfnibov

* * - 5 f (ElE2. * . En) (1 - Enp d&l. 0

In a special case

1 tm Bn” = (m!)” *

The equation

f(f &fhy=O (2.3)

has the solution

O” ( ilk kz (kl)” (hqk = oFn__1 (1, 1, . . . , 1; - hf)= J&‘!. 0 (n fif,. . I. ,

For t = 0 the function J6:,!!., o (0) = 1 (see [d ). Therefore it

follows from (2. l), (2.21, (2.3) that

Therefore

Bn - = J&t!.. 0 (n Tq, 4 +.A -& = IC,!!.. o (n fi,. (2.41

where (see ld)

I&-f!.,o(npf) = ,Fn-,(l, 1, * * *7 1; Al).

Using the last two formulae it is easy to find

Pdting A = lo in (2.4) and using the formulne

Opt-rat ional calculus of I(e.sse 1 operator< 1197

pt 0, o..... 0 (Xl *lifn+l)) = be$)O ,..,, o (x) + i be$‘b ,.... o(x),

I(n) 0. o,.... 0 (x1 +‘fn+l)) = ber:b - i ,..., ,, (2) beipb ,..., o (z).

where (see [cj! )

her!,%,...,

00 (_ Qk (A-)“” (n+tf

O @) = k,. [r (Z/c + I)]” (2k) I ’

03 (_ l)k (_zJ+l) WS1)

be&%,..., o (4 = 2 k_-O Ir (2k + 2)1” (2k i- 1) ! ’

we find

As in the ordinary operational calculus, applying well-known oper- ational methods [31, [71, iSI, to the field of operators %~f, we can

extend the table of valtles of operators consideraMy. For example, differentiating (2.4) with respect to the parameter h and then Putting A = 1 and A = -1 we obtain

where (see id)

1198 V.A. Ditkin and ;2.1’. ~‘ro~fnik#tJ

1

2

3

4

5

6

7

8

9

IO

41

12

13

14

Tables of formulae of the operators B,

In B _ _-.ze-lfB*

0:

-

No.

15

17

18

F(Z”)

43 (B, - i)-

(Table 1 conttnued)

I( t )

i nl! t

m.‘nJ(n-1) m,m.....m 4 ( )

1 P’“I~~~*,,, m! ?A A)

& t- n

imp-1) -%. -‘/x1. . , -% i ) n)/t

tiz t- ?I

Imp-l) -

-‘/,,-‘/* )..., -‘/, ( ) n V’r -

From (2.5) we find

It follows from this that

Multiplying the second of the inequalities (2.6) hy # and summing with respect to II from (1 to CD, we find

i,,TJS B,;;n+i = Bn n &-1-p *

which gives

The operational crrlciilus for the operator

1200 t’.A. nitbin nnft A.P. Pforlnihov

can also be applied to the solution of differential equations. The solu- tions of differential equations of the form

where L(h) = h.” q ~~~~~~ f. - - + a, is a polynomial with constant co-

efficients qi are most simply found. The substitution of the operator

by the operator ,Cn is done according to the formula

it. (t -iv]‘” 5 (t) = BTX (t) - l&x,__, - B2,x,_2 - - - * - &&,

where xk = BXx (t) [I=0 (k = 0, l, 2, . ’ - 9 m - ‘)’

3. 0n the solutions of the differential equation

It was shown in t51 t!:at the operational calculus for the operator

B = $- t -$ is generated hy t!ie equation to w!?ic!

the Cessel equation

ym+;yt -(I - -$)y= 0 (3.1)

for’” = 0 is reduced by the su!xtitution d = 2vc.

Let us consider the 1 inear ordinary differential equation of third order with variable coefficients

y’” + f y” _i_ 1 - 3n12 +;3rnn - 3n? y, $

i * +‘l” + n) (2m -$ n) (n

- 2n)] y = 0 (3.2)

In the theory of third order equations this equation is tl:e analogue of the Bessel equation (3. I).

If m = n = 0 in equation (3.2) then we shall have

(3.3)

Pr~t z = 3+%. Then the preceding equation takes the form

(3.4)

Ey analogy with the way it was done in I:;‘ for the operator

i% =- $ t -$ , starting from a new definition of a bundle and the pro-

perties of solutions of equation (3.4) we can construct an onerational

calculus for the operator 1’ - $ t -$- t -$ (see :!I!, !I01 ) .

:Yithout loss of generality, in equation (3.4) we C3f nut h = J anCJ

consider the equation

t2yS" $ 31y” + y’ -+ y = 0. (3.5)

Let us use the operational method. J,et

Then

(3.7)

M

p e-Pftm $$ dt = (- -&j* [pncp (p)], m>n,

0

cc

s . e+tm $gdl = (- -$j~n[p~c@(p)] -+- (- l)7n-q-(n’“,~1,,l p-m-'y(O) + (I (34

f : (n -- 2) !

(n _ m _ zj! p=m-%J’ (Of _i . * 1 + u2 ! p-m-‘) (O)], lfl <n.

To obtain three linearly independent solutions of equation (3.5) by the operational method it is convenient to consider the equation

1”y”’ + .?tZy” 2 ty’ ” t?y -7 0, (3.9)

which is obtained from (3.5) by mul’tiplication by t.

I\pplying formulae (3.Fjj, (3.7) to the transformation of equation (3.9)

we obtain

After substittItion of the variable according to t2.e formula z = l/i

1202 V./l. Ditbin and A.P. Prodnlkou

equation (3.10) becomes

23qJ”+z(z+I)cp’-cp=o. (3.11)

Equation (3.11) is a special case of an equation depending on the para- meter v:

z3qY + [ 2 (2 + 1) - -5 2 ( 2) +~-[I- (G)]v= 0, (3.12)

when this parameter is equal to zero. If the parameter v is not a whole number, then the functions

where

&r (z, Y) = zJv (2 vi,, (3.13)

G2 (2, v> = ZY” (2 v-3, (3.14)

(p3 (2, Y) := zn, (2 Vi), (3.15)

Jv (z) = mto G; ($‘;:I , (Bessel functions)

Y” (2) = COS (nv) J, (2) - J_, (2)

(sin JCv) ,

I-I” (2) = cos(Jg) ; (- l)m (2/2)2rn r (m + 1 + v/2) r (m + 1 - v/2)

(Poisson function) m=o

form a fundamental system of solutions of equation (3.12). For v = 0 this no longer is the case, since as v - 0 the Poisson function n,,(z) degenerates to the Bessel function ‘Iv(z) and the three linearly inde- pendent solutions (3.13), (3.14), (3.15) of equation (3.12) degenerate into two linearly independent solutions of equation (3.11)

‘~1 (z) = Lb.& (z, v) = hi & (2, 4 = zJ, (2 vi,, (3.16)

92 (2) = !iIy ‘p2 (2, Y) = ZY, (2 Vi). (3.17)

To find the third linearly independent solution of equation (3.11) let us introduce the expression

To apply form in the

lim ~, (‘, ‘) - (p3 (‘, ‘) v-.0

V

1’IlGpital’s rule for the evaluation of the indeterminate last expression we find

opernt ionn 1 cu lcu lus of Resse 1 operrrtnrs 1203

9((z) = $nr (2).

(Jsing the two last equalities we find easily

lim ~~ (‘, ‘) - cPr (2, ‘)

v-0 V = (3.18)

= $

77l=O T In 2 - I# (m:+ l)} = f zY, (2 fi) = $ ‘pz (2).

Consider now the expression

lim 6 0% (2, v) - (93 (2, v)l - f pI (2)

v-4 V (3.19)

In order to use 1’16pital s rule to evaluate the indeterminate form in (3.19) we find

(3.20)

- ; (_.l!mzm+‘~ 2 m! r (v + m -t 1)

q)’ (v f_ m $_ , ),

m 0

1204 V.A. Ditkin and A.P. I’rodnikoa

$ % (2 6) = - $ ~0s (F) 5 r (m + 1 F,,;!$;;; 1 _ ,,,2) (3.21) m -4

+ +iqq ; (- l)mp+e

rn=” r @+1+/q r (m_+_l_-y/2) [II, ww-~/2k-~ (m+l-v/2)1+

+$ cos (F) ; (- l)mp+“fz

r (m+l+v/z) r (vz+l--Y/2) I I$ (m+l +$+9 (m+~---Y/2)p- rn=”

-[W (m -!- 1 - $) + d” (m + 1 + v/2)]).

Further we find

where

lim $ ITI (z* Y) - 6 (z, v)) - $ (p2 (2)

v-+0 V = $ 91 (4 + 93 (a,

‘pS (2) = i (- y;;+ {’ *=lJ

4 ln2z - 9 (m i- 1) In 2 + q2 (m + 1) - $9’ @+I)}.

The function cp,(z) satisfies equation (3.11) and does not depend linearly on the functions rpl(z) and T,(Z). Thus the fundamental system of solutions of equation (3.10) has the form

R(P)= $J”[jy,

qJz (p) = $ Y, A’ i ! VP ’

w$-lp(mi- l)ln$+~2(mS_l)-f~‘(m+l)).

Passing from the we obtain the three

transforms to the originals in the last qualities, linearly independent solutions of equation (3.5) :

ca (- l)m P = m-, (ml)l ;;;r = AJ?, (3 7-a

cplb) = $yo (+-)=~o$&&i {In $ - 29 (m + 1,) + y, (t) =

Operationa E co tcu lus of Resse 1 operators 1205

(In2 t - 69 (m + 1) In t + 9qz (m + 1) - 39,’ (m + 1)).

4. The operator T = $ t&t$

Now let L, denote the set of all functions f(t) defined on the half- line 0 <t < m and Lebesgue integrable on any finite interval (0, $4) and satisfying the condition

0 0 0

for any t0 > 0. Let “4 denote the set of all functions of the form

where f(t) is any function of .$. and I: is any constant. The sets k, and MI. are the same as the sets LB and hfB introduced in Section 1, with

n = 3. Therefore, Wfr is a ringnof typenhf, where the product is defined by the formula

If

and

1206 V.A. nit&in and A.P. Prodnikov

then we shall have

Ve denote the extended ring by a~. For the operator l/t we introduce the notation l/t = T. Then for the inverse operator T’ = l/T we shall have l/T = t . Therefore

or

If F(t)ehl~ and F(0) = 0, then

TF(t) = $ t $ t $ F(t) = PF" (t) + 3@"(t) + F'(t). (4.3)

It follows from (4.3) that

-J+/(t) =I &$t -r)“dr~(1-2z)%~f(.%yI)(1 --?J)“dy. (4.4)

0 0 0

In a special case we have

For t = 0 one of the linearly independ~t solutions of equation (3.4),

J7J3$%t)=,F,(1,1, - ht), takes the value J(o’I)o (0) = 1. merefore

it follows from (3.4), (4.2). (4.3) that

This gives

(4.5)

Operat iona 1 co lcu lus of Cesse 1 operators 1207

where

Ip,(z) = ,&[I, 1; (+)3J.

Using formulae (4.5) we easily obtain

Putting A = io in (4.5) we find easily

(4.6)

(4.7)

= be$,, (3 a),

UT TI+o’ 2

O3 (- l)k (at)*+1

K=O i(2k + 1) II* = beit’, (3 fi).

BY analogy with the way it was done for the operator B = $- t $ ,

it is easy to obtain for the operator T = $ t -$ t $

T

(T + i)“+’ (4.8)

T

(T - i)“+’ (4.9)

Here

(z/3)2” CU) = pQ&+i) OFS[n+l,n+~,(~)‘].

Further

T (G) = $$$ = o bert), (3 Tii$,

T(& -l)=-+$p-~ (

UT T'+oZ =- 1

o beit’,-, (37s).

It follows from this that

1208 V.A. Ditkin and A.P. Pro~nikov

Multiplying equality (4.9) by pn and summing for n from 0 to m we find

which gives

The operational calculus for the operator Ip =-g+--$ can he

constructed by starting from the corresponding integral transformation. the integral transform The analogue of a Laplace transform here will be

f’(T) = $ (t).~(~t)~t,

0

where

is the solution of equation (3.4) for h = 1. The operational calculus

d d d for the operator T = dt t dt t 3 can also be applied to the solution

of differential equations.

Let

q-&t $ t $)cErt, =f($ (4.10)

where L(h) = h* + aIAn-’ + - - - + a, is a polynomial with constant co-

efficients ni. The operator Gt% t-$- in equation (4.10) is replaced

by the operator T according to the formula

where

5k = Tk I (t) It=0 (k=O, 1, 2, . . . , n- 1;.

f$erat iona 1 calculus of Besse 1 operators 1209

TABLE 2.

Tables of foraulae of the operators T

F 0’)

1

1 --ii X

T Xa + a8

T l--l9

T* Ta - a=

T* T’+aa

TV’+4 (T + a)% - az

T T~faT”-+a*T +aa

TZ TS+aTa+a4T+as

TS + aTZ + aaT + as

T T&-as

f (0

I

1210 V.?I. Ditkin and A.!‘. Prodnibov

16

18

19

20

21

23

25

26

27

28

{Table 2 cant inuad)

F CO f (I)

T2 Ta--a4

TO

TP-a4

T

(T + l)m+l

T

(T - Qm+l

T

12 _ 1)v [G” + 1)” - U’- I)“1

T

(T + I)“+l

1 ?;;;; =

-l/T

I -

I

N,

29

30

31

32

33

34

35

Opernt ionn 1 calculus of Resse I operators 1211

(Table 2 continued)

F (T) t (0

--- -~.-_--

r [sin (T) ci (T) - cos (T) si (T)] ber (2 r/t)

r [ cos (T) ci (T)+sin (T) si (T)] - bei (2 1/i)

f?; cT erfr, (VT)

TeTKo (7’)

T [Ho (T) - Yo G”)f $ Ja( y-Z) IO (Jfr2)

1 -TeTIK~(T)--Ko(V 1 si (2 I/t)

For example, let us find the solution of the equation

( $4 d)2 tdt s(t) + J:(1) --: (!

with the condition

s(o) = I, Tx(t)jt.zn .ez ft%“’ (t) + &x"(t) + s'(f)]~, = 1.

After replacing the operator have

-34 t f by the operator 7’ we will

T2x (t) - T - T2 -/- z(I) -: 0,

which gives

Trnnslatcd by R. Feinstein

1212 V.A. !)itkin am! .I.?. i’rodnikov

REFERENCES

1.

2.

3.

4.

5.

6.

7.

a.

9.

Delsarte, J., Acta Flathematica, 69, NOS. 3-4, 259-317, 1936.

Mikusinskii, Ya., Operational Calculus (Operatornoe ischislenie).

Izd-vo in. lit. Moscow, 1956.

Ditkin, V.A. and Prudnikov, A.P., Integral Transforms and Operational

Calculus (Integral’nye preobrazovaniya i operat+onnoe ischis-

lenie). Fizmatgiz, Moscow, 1961.

Mikisinski, J.G. and Ryll-Nardzewski, C., Studia Meth., 13, 62-66,

1953.

Ditkin. B.A., Dokl. Akad. Nauk SSSR, 116, 15-17, 1957.

Delerue, P., Sur le calcus symbolique i n variables. Thesis, Mont-

pellier, 1951.

Ditkin, V.A. and Prudnikov, A.P., Operational Calculus in two Vari-

ables and its Applications (Operatsionnoe ischislenie po dvum

peremennym i ego prilozheniya). Fizmatgiz, Moscow, 1958.

Efros, A.M. and Danilevskii, A.M., Operational Calculus and Contour

Integrals (Operatsionnoe ischislenie i- knotyrnye integraly). Cos.

nauchno- tekh. izd-uo Ukrain. Kharkov, 1937.

Prudnikov, A. P., Dokl. Akad. Nauk SSSR,, 142, No. 4. 794-797, 1962.

10. Prudnikov, A. P., Dokl. Akad. Nauk SSSR, 144. No. 1, 56-57. 1962.