Fractional operators and special functions. I. Bessel functions

Fractional operators and special functions. I. Bessel functionsLoyal Durand Citation: Journal of Mathematical Physics 44, 2250 (2003); doi: 10.1063/1.1561593 View online: http://dx.doi.org/10.1063/1.1561593 View Table of Contents: http://scitation.aip.org/content/aip/journal/jmp/44/5?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Legendretype Special Functions Defined by Fractional Order Rodrigues Formula AIP Conf. Proc. 1301, 644 (2010); 10.1063/1.3526666 Partial fractions expansions and identities for products of Bessel functions J. Math. Phys. 46, 043509 (2005); 10.1063/1.1866222 Associated Bessel functions and the discrete approximation of the free-particle time evolution operator incylindrical coordinates J. Math. Phys. 45, 1988 (2004); 10.1063/1.1695601 Fractional operators and special functions. II. Legendre functions J. Math. Phys. 44, 2266 (2003); 10.1063/1.1561594 Recursive construction for a class of radial functions. I. Ordinary space J. Math. Phys. 43, 2707 (2002); 10.1063/1.1463709

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Fractional operators and special functions. I. Besselfunctions

Loyal Duranda)

Department of Physics, University of Wisconsin, Madison, Wisconsin 53706

~Received 6 December 2002; accepted 3 January 2003!

Most of the special functions of mathematical physics are connected with the rep-resentation of Lie groups. The action of elementsD of the associated Lie algebrasas linear differential operators gives relations among the functions in a class, forexample, their differential recurrence relations. In this paper, we define fractionalgeneralizationsDm of these operators in the context of Lie theory, determine theirformal properties, and illustrate their use in obtaining interesting relations amongthe functions. We restrict our attention here to the Euclidean group E~2! and theBessel functions. We show that the two-variable fractional operator relations leaddirectly to integral representations for the Bessel functions, reproduce known frac-tional integrals for those functions when reduced to one variable, and contribute toa coherent understanding of the connection of many properties of the functions tothe underlying group structure. We extend the analysis to the associated Legendrefunctions in a following paper. ©2003 American Institute of Physics.@DOI: 10.1063/1.1561593#

I. INTRODUCTION

Most of the classical special functions are connected with the representation of Lie groups,1–4

and appear as factors in multivariable functions on which the action of an associated Lie algebrais realized by linear differential operators. Many of the properties of the special functions areeasily understood in this context. For example, the differential equations for the special functionsare connected with the Casimir operator of the associated groups. The actions of appropriateelementsD of the Lie algebra lead, when reduced to a single variable, to the standard differentialrecurrence relations for the functions, while the action of group elementse2tD can be interpretedin terms of generalized generating functions when expressed using a Taylor series expansion in thegroup parametert. Numerous examples are given in Refs. 1 and 3. In the present paper, we willdefine fractional generalizationsDm of theD ’s in the context of Lie theory, determine their formalproperties, and illustrate their usefulness in obtaining further interesting relations among the func-tions, including integral representations for the functions. Most of the specific results have beenderived historically in other ways, but are unified here in a group setting.

Two examples of fractional operators in a single variable are provided by the fractionalintegrals of Riemann and Weyl~Ref. 5, Chap. 13!. These give a useful way of changing the indices~degree or order! of the classical orthogonal functions~Jacobi, Gegenbauer, Legendre, Laguerre,Bessel, and Hermite functions!. An example is Sonine’s first integral for the Bessel functions,6

12.11.~1!,

xn1mJn1m~x!5xm

2m21G~m!E

0

p/2

Jn~x sinu!cos2m21 u sinn11 u du ~1!

51

2m21G~m!E

0

x

tn11Jn~ t !~x22t2!m21 dt. ~2!

a!Electronic mail: [email protected]

JOURNAL OF MATHEMATICAL PHYSICS VOLUME 44, NUMBER 5 MAY 2003

22500022-2488/2003/44(5)/2250/16/$20.00 © 2003 American Institute of Physics


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The expression on the second line is equivalent to the Riemann fractional integralRmxn/2Jn(2Ax),where the integral operatorRm is defined in general~Ref. 5, Chap. 13! as

~Ra f !~x!51

G~a!E

0

x

f ~ t !~x2t !a21 dt. ~3!

Thus, with the replacement oft by 2At andx by 2x, ~2! becomes

x(n1m)/2Jn1m~2Ax!51

G~m!E

0

x

tn/2Jn~2At !~x2t !m21 dt5Rmxn/2Jn~2Ax!. ~4!

A number of similar results are known for other special functions, for example,

2l2m~12x2!2(l2m)/2Pnl2m~x!5Rm2l~12x2!2l/2Pn

l~x! ~5!

for the associated Legendre functions with Rel,1, Rem.0 @Ref. 5, 13.1~54!#. Askey ~Ref. 7,Chap. 3! summarizes a number of results and gives some applications.

Other results are known with respect to the Weyl fractional integralWm ~Ref. 5, Chap. 13!defined by

~Wa f !~x!51

G~a!E

x

`

f ~ t !~ t2x!a21 dt. ~6!

Thus, from Ref. 5, 13.2~59!,

x2(n2m)/2Kn2m~2Ax!5Wmx2nKn~2Ax!, ~7!

whereKn is the hyperbolic Bessel or MacDonald function.The simplicity of the results noted, and of many similar results,7 is striking. The effect of the

fractional integration is simply to change the indices on the special functions, while retaining theoriginal functional form. There does not appear to be a systematic approach to the derivation ofthese results in the literature. Their form suggests that they must be associated with fractionalgeneralizations of the stepping operators in the associated Lie algebra. In particular, the differen-tial recurrence relations for the special functions are schematically of the formDFa,...

5cFa61,... , whereD is a linear differential operator and the indicesa label the functions in arealization of the algebra. This suggests that shifts of the indices by arbitrary amounts could beeffected using fractional operatorsDm defined in analogy to the single-variable fractional deriva-tives defined in Ref. 5. This is the case, as we will see. The above-given fractional integrals arerelated, and simply give the action of the inverse multivariable operatorsD2m when reduced to asingle variable.

We will define the fractional operatorsDm in the context of Lie theory and explore theirgeneral properties in Sec. II. We will then apply the results in a number of group settings in thisand following papers to obtain generalized fractional-integral-type relations of the formFa1m,...

5NDmFa,... for the special functions. Some are apparently new. We find that, with appropriatechoices for the input functions, the fractional relations lead directly to known integral representa-tions for the special functions, providing a group-theoretical setting for the latter.

In the present paper, we will restrict our attention to the development of our methods, and toapplications to the Bessel functions. Our treatment is not exhaustive in either the theory or theapplications considered.

II. FRACTIONAL OPERATORS

We will suppose that we have a Lie algebra which corresponds to one of the classical Liegroups, and is realized by the action of a set of linear differential operators$D(w,]w)% in a

2251J. Math. Phys., Vol. 44, No. 5, May 2003 Fractional operators and special functions. I.


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collection of variablesw acting on an appropriate set of functions. The exponentialse2tD definedby Taylor series expansion in the group parametert are elements of the Lie group taken to act onan appropriate class of functionsF. We will assume that the group actione2tDF can be definedfor all t, and will define a Weyl-type fractional operatorDW

m as an integral over group elements by

DWm ~w,]w!F~w!5

1

2p ieipmG~m11!E

CW

dte2tD(w,]w)

tm11 F~w!. ~8!

The contourCW5(`,01,`) in the complext plane runs in from infinity, circlest50 in thepositive sense, and runs back to infinity. To define phases, we take the integrand as cut along thepositive real axis with the phase oft taken as zero on the upper edge of the cut. The direction ofthe contour at infinity must be such that the integral converges. The above-given expression wouldbe an identity forD a positive constant. Here, however,D(w,]w) is an operator which acts onfunctionsF of the collection of variablesw, and the existence of the integral depends on thefunctions as well as the contour.

Alternatively,DWm can be defined as

DWm F5

1

G~2m1n!DnE

0

`

dte2tD

tm2n11 F, ~9!

where Rem,n and end-point terms are assumed to vanish in the partial integrations which con-nect the two expressions.

It is straightforward using this expression to show that the fractional operators have theexpected algebraic properties. Thus, for Rem,0, Ren,0, and Re(m1n),0,

DWm DW

n F51

G~2m!G~2n!E

0

`

dtE0

`

du1

tm11

1

un11 e2(t1u)DF

51

G~2m!G~2n!E

0

`

dvE0

vdt

1

tm11

1

~v2t !n11 e2vDF

51

G~2m!G~2n!E

0

`

dve2vD

vm1n11 • E0

1

dt8 t82m21~12t8!2n21

51

G~2m2n!E

0

`

dve2vD

vm1n11 5DWm1nF. ~10!

Exponents therefore add as we would expect, and the fractional operators of different orderscommute,

DWm DW

n 5DWn DW

m 5DWm1n , @DW

m ,DWn #50. ~11!

The result extends through~9! to generalm,n for which the fractional operators are defined. Byconverting the integral in~10! back to a contour integral before taking the limitn→2m, we findalso thatDW

m DW2m51 where1 is the unit operator, soDW

2m is the inverse ofDWm as implied by the

group operations.The fractional operatorDW

m can also be defined in terms of the action of a generalized Weylfractional integralW2m in the parameterx on the group operatore2xD(w,]w). We will defineW2m

for generalm as

W2m f ~x!51

2p ieipmG~m11!E

Cx

dtf ~ t !

~ t2x!m11 dt, ~12!

2252 J. Math. Phys., Vol. 44, No. 5, May 2003 Loyal Durand


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whereCx is the contour ( ,x1,`). This definition is equivalent to~8! for Rem,0. The action ofW2m on f is just that of a fractional derivative,

~W2m f !~x!5~2d/dx!m f ~x!, ~13!

a result which is obvious form an integer so that the integration contour can be closed. In general,W2m gives the inverse ofWm thought of as a repeated integral,W2mWm51.

DWm can now be defined formally through the action of the fractional derivative (2d/dx)m on

e2xD, (2d/dx)me2Dx5Dme2Dx. Multiplication by exD then givesDWm F5exD(W2me2xDF ).

This relation is easily checked by usingf (t)5e2tDF in ~12! and changing the integration variablefrom t to t2x. We find that

DWm F5exD~W2me2xD!F5

1

2p ieipmG~m11!E

CW

dte2tD

tm11 F ~14!

in agreement with~8!. That is,DWm F5exD W2m(e2xDF ) whereW2m acts on the group parameter

x and D(w,]w) acts onF(w). The integrals in~8! and ~14! can also be identified directly as(2d/dx)me2xDux505DW

m . The inverse of the fractional operatorDWm F is DW

2mF5exD Wm(e2xDF ).

We can define a second Riemann-type fractional operator by replacing the Weyl fractionalintegral by a Riemann fractional integral and noting the correspondence ofR2a to (d/dx)a,R2a f (x)5(d/dx)a f (x). Thus, takingf (t)5etDF and a5n2Rem.0 in ~3! and following theabove-given construction, we find

DRmF5e2xDDn~Rn2mexD!F, 0,n2Rem. ~15!

By changing the integration variable fromt to x2t in ~3!, we then obtain the analog of~9!,

~DRmF !~x!5

1

G~2m1n!DnE

0

x(w)

dte2tD

tm2n11 F, ~16!

where we have noted the dependence of the final result on the valuex(w) of the group parametert at the end point of the integration. As indicated, this will depend on the values of the variablesw in F.

By going to a contour integral to handle the possible singularity at the lower limit of integra-tion, we can writeDR

mF in the more general form

DRmF5

1

2p ieipmG~m11!E

CR

dte2tD

tm11 F, ~17!

whereCR is the contourCR5(x(w),01,x(w)).As we will see explicitly in later applications, the end pointx(w) of the contour must be

chosen such thatDmF satisfies a differential equation determined by the Casimir operators of theLie algebra. This will require that a differential expression related toe2tDF vanishes fort5x(w) for the given values of the variablesw in F @see, for example,~85!#.

The product of two Riemann fractional operators is given in the simple case Rem, Ren,0 by

DRm~DR

n F !~x!51

G~2m!G~2n!E

0

x

dtE0

x

due2(t1u)D

tm11un11 u~x2t2u!. ~18!

Dm(DnF ) will satisfy the expected differential equation forDmG providedt1u5x on the bound-ary of the region of integration, a condition is enforced in~18! by the unit step functionu(z),u(z)51 for z.0 andu(z)50 for z,0. For an explicit example, see Sec. III D 1. The integral canbe evaluated by shifting tov5t1u as a new integration variable and identifying the remaining



138.38.171.231 On: Wed, 26 Nov 2014 13:41:43

integral with a beta function as in~10!. The result is equal toDRm1nF. We therefore obtain the

multiplication relationDRmDR

n 5DRn DR

m5DRm1n derived earlier for the Weyl fractional operators.

This can be generalized to the contour integral representation~17!.Which expression forDm is appropriate in a particular setting, Weyl or Riemann, will depend

on D andF. We will therefore simply denote the fractional operator asDm for formal purposes,and only specify the expression to be used in particular applications. The key restrictions will bethe existence of a finite value of the group parametert5x(w) such thate2xDF50 in the Riemanncase, and convergence of the integral fort→` in the Weyl case.

III. BESSEL FUNCTIONS AND E „2…

A. Algebraic considerations

As a first application of the fractional operators, we will consider the Bessel functions whichwe will denote generically asZn(x). Bessel functions appear naturally in representations of E~2!,the Euclidean group in two dimensions, and of E~1,1!, the Poincare´ group in two dimensions.1,2

Both groups are real forms of SO(2,C), and the two are related to each other through the Weylunitarity trick.8 Since we are not concerned with unitary representations of the groups, it will besufficient for our purposes to consider only the algebra of E~2!.

The Lie algebra of E~2! is generated by three operatorsP1 , P2 , J3 with the Lie products orcommutation relations

@P1 ,P2#50, @J3 ,P1#5P2 , @J3 ,P2#52P1 . ~19!

There is one invariant operator, namelyP121P2

2, which commutes with all the generators.The algebra can be realized by the action of differential operators on functionsf of the

coordinates (x1 , x2) in the Euclidean plane.P1 andP2 correspond to the translation operators

P15]1 , P25]2 ~20!

andJ3 , to a rotation in the plane,

J352x1]21x2]1 . ~21!

The condition that the invariant operatorP121P2

2 be constant on the functionsf gives theHelmholtz equation (P1

21P22) f 52k2f . In polar coordinatesx, f this becomes the differential

equation

~P121P2

21k2! f 5F ]2

]x2 11

x

]

]x1

1

x2

]2

]f2 1k2G f 50. ~22!

We can takek251 by a scaling of the coordinates, and will do so. The rotation operatorJ3

52]f commutes with the Helmholtz operator and may also be taken to have a constant value2 in on the functions. The functionsf in this realization of E~2! are then of the form

f n~x,f!5einfZn~x!, ~P121P2

211! f 50, J3f n52 in f n ~23!

and involve Bessel functionsZn of ordern.It is useful to change from the anti-Hermitian operatorJ3 to the Hermitian operatoriJ3 , and

to introduce operators

P152P12 iP2 , P25P12 iP2 ~24!

with the commutation relations

@P1 ,P2#50, @ iJ3 ,P6#56P6 . ~25!



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The last relations imply that iff n is a solution of the Helmholtz equation with the indexn, thenP6 f n is a solution with indexn61,

iJ3~P6 f n!5P6~ iJ361! f n5~n61!~P6 f n!. ~26!

P6 therefore act as stepping operators on the index.The operators are given explicitly by

P152eifS ]

]x1

i

x

]

]f D52t]x1t2

x] t , ~27!

P25e2 ifS ]

]x2

i

x

]

]f D51

t]x1

1

x] t , ~28!

wheret5eif. In terms of that variable,iJ35t] t . The Helmholtz operator is simplyP1P211.From ~26!, the action ofP6 on the functionsf n(x,t)5tnZn(x) must give constant multiples

of tn61Zn61(x). The constants of proportionality for the different Bessel functions are easilydetermined to be unity by using the behavior of the functions forx→0, . We therefore have thestepping relations

P6tnZn~x!5tn61Zn61~x!, ~29!

which reduce to

S 7d

dx1

n

xDZn~x!5Zn61~x! ~30!

once thet dependence is factored out. The latter are just the differential recurrence relations forthe Bessel functions~Ref. 9, 7.2.8!.

The relations in~29! suggest that

P6m tnZn~x!5tn6mZn6m~x! ~31!

for P6m appropriately defined fractional operators such as the Weyl operators

P6m 5

1

2p ieipmG~m11!E

CW

due2uP6

um11 . ~32!

It is easily established that these operators have the expected properties. First,P6m commute

with the Helmholtz operatorP1P211, so transform solutions of the Helmholtz equation tosolutions. Further, from

@ iJ3 ,P6n #56nP6

n , ~33!

we find that

@ iJ3 ,e2uP6#56 (n50

`~2u!n

n!nP6

n ~34!

56ud

due2uP6, ~35!

hence, after a partial integration in~32!, that



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@ iJ3 ,P6m #56mP6

m . ~36!

The action ofP6m on a solutionf n therefore gives another solution with the indexn changed to

n6m,

iJ3~P6m f n!5P6

m ~ iJ36m! f n5~n6m!~P6m f !. ~37!

This relation does not show directly thatP6m f n5 f n6m , but only thatP6

m f n is at most a linearcombination of the two independent solutions of the Helmholtz equation with ordersn6m. If theindependent solutions are taken as the Hankel functions, the observation that the operatorsP6

m donot change the distinct asymptotic behaviors of those functions foruxu→` shows, in fact, thatP6

m f n5N(n,m) f n6m . The constant of proportionality will be found later by direct calculation tobe unity, as in~52!, establishing the validity of~31!.

We can also define a fractional operator (iJ3)l, and find after a brief calculation using theanalog of~32! that

~ iJ3!l f n5nl f n . ~38!

( iJ3)l again satisfies the multiplication rule, (iJ3)l( iJ3)m5( iJ3)l1m.The formal algebraic structure is completed by

~ iJ3!lP6m 5P6

m ~ iJ36m!l, P6m ~ iJ3!l5~ iJ37m!lP6

m . ~39!

These can be derived using the Baker–Hausdorff expansion ofeABe2A as a series ofn-foldcommutators,

eABe2A5B1 (n51

`1

n!@A,@A,...@A,B#...##. ~40!

Thus, choosingeA as the exponential in the definition of (iJ3)l, eA5e2 i tJ3, B asP6m , and using

~36! to evaluate the repeated commutators, we find that

e2 i tJ3P6m 5~e2 i tJ3P6

m eitJ3! e2 i tJ35P6m 1 (

n51

`~2t !n

n!@ iJ3 ,iJ3 ,...@ iJ3 ,P6

m #...#] e2 i tJ3

5P6m e2t( iJ36m). ~41!

The first of the relations~39! then follows upon integration using the analog of~32!. Applicationof this operator to a solutionf n of the Helmholtz equation gives

~ iJ3!lP6m f n5P6

m ~ iJ36m!l f n5~n6m!lP6m f n ~42!

The second of the relations~39! can be derived similarly. The complete algebraic structuredefined by (iJ3)l, P6

m , the multiplication rules, and~39! is infinite, and has not been investigatedexcept as applied to solutions of the Helmholtz equation.

B. Action of the group operators

The action of the exponential operatorse2uP65e6uP11 iuP2) is easily determined and wellknown. P1 and P2 commute, and the exponentialseaP1 and eaP2 induce translations of thecoordinatesx1 , x2 with eaP1x15x11a andeaP2x25x21a. Thus, acting on functions analytic inthe neighborhood of (x1 ,x2),

e2uP1F~x1 ,x2!5euP1eiuP2F~x1 ,x2!5F~x11u,x21 iu !. ~43!



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Applying this result to the functionsf n5einfZn(x) written in rectangular coordinates, we find that

e2uP1 f n5eu(P11 iP2)S x11 ix2

x12 ix2D n/2

Zn~Ax121x2

2 !5tnxn~x212uxt!2n/2Zn~Ax212uxt !, ~44!

wherex5Ax121x2

2 and t5eif5A(x11 ix2)/x. A similar calculation gives

e2uP2 f n5S t

xD nS x222ux

t D n/2

ZnSAx222ux

t D . ~45!

We can also calculate directly in polar coordinates, a method which will be useful later. Thus,noting thatP1tnxn5(t/x)(t] t2x]x)t

nxn50 and using~30!, we find that

P1n tnZn~x!5~22!ntn1nxn1nS d

dx2D n

~x2nZn~x!!. ~46!

The formal Taylor series expansion ofe2uP1 then gives

e2uP1tnZn~x!5 (n50

`~2u!n

n!~xt!n1nS d

dr 2D n

~r 2nZn~r !!ur 5x

5tnxn expS 2uxtd

dwD ~w2n/2Zn~Aw!!uw5x2

5tnxn~x212uxt!2n/2Zn~Ax212uxt !, ~47!

where we have identified the exponential in the penultimate line as a translation operator. Theresult agrees with~44!. A similar calculation fore2uP2 reproduces~45!.

Direct evaluations ofe2uP6 f n using the Taylor series for the exponentials and the relationsP6

n tnZn(x)5tn6nZn6n(x), ~29!, give the generating functions

tnxn~x212uxt!2n/2Zn~Ax212uxt !5 (n50

`~2u!n

n!tn1nZn1n~x! ~48!

and

S t

xD nS x222ux

t D n/2

ZnSAx222ux

t D 5 (n50

`~2u!n

n!tn2nZn2n~x!. ~49!

These equations give generalizations of Lommel’s expansions for the Bessel functions~Ref. 6,Sec. 5.22!. Thus, takingx5Az, ut5h/2x5h/2Az in ~48!, and choosingZn as the ordinary Besselfunction Jn , we obtain@Ref. 6, 5.22~1!#

~z1h!2n/2Jn~Az1h!5 (n50

` ~2 12 h!n

n!z2(n1n)/2Jn1n~Az!. ~50!

Similarly, for x5Az andu/t52h/2Az, ~49! gives @Ref. 6, 5.22~2!#

~z1h!n/2Jn~Az1h!5 (n50

` ~ 12 h!n

n!z(n2n)/2Jn2n~Az!. ~51!



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The remaining Lommel-type formulas in Ref. 6, Sec. 5.22, follow from~48! and~49! for differentchoices ofZn . The present development provides a group-theoretical derivation of these results.See also Weisner.10 Note the series~48! converges foruuu,ux/2tu, and that in~49!, for uuu,uxt/2u, that is, for sufficiently small values of the group parameteru.

C. Weyl-type relations for Bessel functions

1. Relations using P ¿m

The action of the Weyl-type operatorsP1m on the Bessel functions is given by~8! and ~44!,

P1m tnZn~x!5N~n,m!tn1mZn1m8 ~x!

51

2p itnxneipmG~m11!E

CW

du

um11 ~x212uxt!2n/2Zn~Ax212uxt !. ~52!

CW is a contour ( ,01,`) in the complexu plane with the direction of approach toto be takensuch that the integral converges. This will depend on the functionZn considered.

Proceeding formally, we can extract the expected factortn1m from the integral by the changeof variablev52uxt. We will also replacex by Az, with the result

N~n,m!x2(n1m)/2Zn1m8 ~Az!51

2p i2meipmG~m11!E

CW

dvvm11 ~v1x!2n/2Zn~Av1z !. ~53!

This result can also be obtained directly from the differential recurrence relations~30! by replacingx by Az, rewriting the resulting relation forP1 in the form

22d

dz~z2n/2Zn~Az!!5z2(n11)/2Zn11~Az!, ~54!

and determining the Weyl action of (22 d/dz)m on z2n/2Zn(Az).The functionz2l/2Zl(Az) satisfies the differential equation~Ref. 11, 9.1.53!

S d2

dz2 1l11

z

d

dz1

1

4zD z2l/2Zl~Az!50. ~55!

Applying this operator withl5n1m to the integral in~53!, converting the derivatives withrespect toz to derivatives with respect tov, and using the differential equation forl5n toeliminate the derivative-free term proportional to 1/4z on the right-hand side, we find that theresult vanishes provided

ECW

dvd

dv H 1

vm

d

dv@~v1z!2n/2Zn~Av1z!#J 50. ~56!

That is, the integral in~53! gives a Bessel function or combination of functions with argumentAzand ordern1m multiplied byz2(n1m)/2 provided the function in curly braces vanishes at the endpoints of the integration.

WhenZn is the Hankel functionHn(1) , the condition in~56! is satisfied for contours that run

to ` in the upper half plane, avoiding the possible singularity atv52z on the right. It also holdsfor a contour along the positive real axis for Re(m11

2n134).0. In either case, an asymptotic

argument shows that the Bessel functionZn1m8 given by the integral is in factHn1m(1) (Az) with

coefficientN(n,m)51 as expected. Thus,



138.38.171.231 On: Wed, 26 Nov 2014 13:41:43

z2(n1m)/2Hn1m(1) ~Az!5

1

2p i2meipmG~m11!E

(`ei e,01,`ei e)

dvvm11 ~v1z!2n/2Hn

(1)~Av1z !,

~57!

e.0. Tracing the calculation back, we find that the original expression~52! holds for Hn(1) for

contours with 0<arg (xtu)<2p as uu u→`.A shift of the integration variable brings~57! to the form of a~generalized! Weyl fractional

integral,

z2(n1m)/2Hn1m(1) ~Az!5

1

2p i2meipmG~m11!E

(`,z1,`)

dv~v2z!m11 v2n/2Hn

(1)~Av !. ~58!

The contour can be collapsed for Rem,0, and~58! reduces to the known fractional [email protected], 13.2~45!#. The latter can be written in the present notation asz2(n2m)/2Hn2m

(1) (Az)5P1

2mz2nHn(1)(Az), Rem.0.

Similar considerations for the choiceZn5Hn(2) in ~52! show that that result holds for 0

>arg(xtu).2p, and that

z2(n1m)/2Hn1m(2) ~Az!5

1

2p i2meipmG~m11!E

CW

dvvm11 ~v1z!2n/2Hn

(2)~Av1z !, ~59!

wherev runs to` in the lower half plane, avoiding the possible singularity atv52z on the right.The result also holds for a contour along the positive real axis for Re(m11

2n134).0.

Combinations ofHn(1) andHn

(2) give the ordinary Bessel functions and the relations

z2(n1m)/2Jn1m~Az!51

2p i2meipmG~m11!E

CW

dvvm11 ~v1z!2n/2Jn~Av1z !, ~60!

z2(n1m)/2Yn1m~Az!51

2p i2meipmG~m11!E

CW

dvvm11 ~v1z!2n/2Yn~Av1z ! ~61!

for Re(m112n13

4).0. The contours in these cases must be taken parallel to the real axis forv→`. The results reduce to the known fractional integrals@Ref. 5, 13.2~34! and 13.2~40!# forRem,0.

If we increase the phase ofz by p and simultaneously rotate the contourCW by p in thepositive sense in expression~57!, the substitutionsz5eipx, v5eipu restore the original contourwhile replacingAv1x by eip/2Au1x. The definition of the MacDonald functionKn in terms ofthe Hankel functionHn

(1) ,

Kn~x!5ip

2eipn/2Hn

(1)~eip/2x!, ~62!

then gives

x2(n1m)/2Kn1m~Ax!51

2p i2meipmG~m11!E

CW

du

um11 ~u1x!2n/2Kn~Au1x!, ~63!

or, for Rem,0,



138.38.171.231 On: Wed, 26 Nov 2014 13:41:43

x2(n1m)/2Kn1m~Ax!52m1

G~2m!E

0

` du

um11 ~u1x!2n/2Kn~Au1x!

52m1

G~2m!E

x

` dt

~ t2x!m11 Kn~At ! ~64!

in agreement with~7! or Ref. 5, 13.2~59!.

2. Weyl-type relations from P Àm

The action of the Weyl-type operatorsP2m on the Bessel functions is given by~8! and ~45!,

P2m tnZn~x!5tn2mZn2m~x!5

1

2p i S t

xD n

eipmG~m11!ECW

du

um11 S x222ux

t D n/2

ZnSAx222ux

t D .

~65!

CW is again a contour ( ,01,`) in the complexu plane with the direction of approach toto betaken such that the integral converges. We will scale out thet dependence through the substitu-tions v52ux/t andx5Az, and work with the reduced expression

z(n2m)/2Zn2m~Az!51

2p i2meipmG~m11!E

CW

dvvm11 ~z2v !n/2Zn~Az2v !. ~66!

We will suppose initially that argz.0. It is then possible for the choiceZn5Hn(1) to rotate the

integration contour into the lower halfv plane. Then withv replaced bye2 ipv and z2v byeip(v1z),

z(n2m)/2Hn2m(1) ~Az!5

1

2p i2me2p imG~m11!E

CW

dvvm11 ~v1z!n/2Hn

(1)~Av1z!, ~67!

where CW is a contour ( ,01,`) in the new variablev and 2p,argz,p. By choosingZn

5Hn(2) and Imz,0 and rotating in the opposite sense, we obtain the second relation

z(n2m)/2Hn2m(2) ~Az!5

1

2p i2mG~m11!E

CW

dvvm11 ~v1z!n/2Hn

(2)~Av1z!, ~68!

also valid for2p,argz,p. These results can also be obtained by consideringP12mtnZn .

For Rem,0, the contours can be collapsed, and

z(n2m)/2Hn2m(1,2)~Az!5

2m

G~2m!e6 ipmE

0

` dvvm11 ~v1z!n/2Hn

(1,2)~Av1z!, ~69!

where Imv→6` for H (1) andH (2). By considering the limiting behavior for Imv→0 and com-bining the two functions, we obtain

2m

G~2m!E

0

` dvvm11 ~v1z!n/2Jn~Av1z!5z(n2m)/2@cospmJn2m~Az!1sinpmYn2m~Az!#,

~70!

2m

G~2m!E

0

` dvvm11 ~v1z!n/2Yn~Av1z!5z(n2m)/2@cospmYn2m~Az!2sinpmJn2m~Az!#.

~71!



138.38.171.231 On: Wed, 26 Nov 2014 13:41:43

These are equivalent to@Ref. 5, 13.2~35! and 13.2~39!# and are valid only for12 Ren234,Rem

,0, with Imv→0 for Rev→`.

3. Weyl-type integral representations for Bessel functions

We can use the above-given results to obtain integral representations for the Bessel functions.We begin with the observations thattmZm(x)5P1

m21/2t1/2Z1/2(x), and thatH1/2(1)(x) andH1/2

(2)(x) areelementary functions,

H1/2(1)~x!5

1

i S 2

pxD 1/2

eix, H1/2(2)~x!52

1

i S 2

pxD 1/2

e2 ix. ~72!

The action ofP1m21/2 can be reduced as above, and we will begin with the expression in~53!. This

gives

x2m/2Hm(1)~Ax!5

1

2p i2m2 1/2ei (m2 1/2)pGS m1

1

2D ECW

dvvm1 1/2~v1x!21/4H1/2

(1)~Av1x!

521

2p i

2m

ApeipmGS m1

1

2D ECW

dvvm1 1/2~v1x!21/2eiAv1x. ~73!

Replacingx by x2, letting v5x2(t221), and removing a common factor ofx2m, we obtain

Hm(1)~x!52

1

2p i

2

ApS 2

xD m

eipmGS m11

2D E(`,11,`)

dt

~ t221!m1 1/2eixt. ~74!

This holds for general values ofm provided Imxt→` for utu→`, and for xt→1` for Rem.1

20. The contour can be collapsed for Rem,12 giving the generalized Mehler–Sonine integral

representation forH (1)(x) @Ref. 6, 6.13~1!#,

Hm(1)~x!52

2i

ApS 2

xD m 1

G~ 12 2m!

E1

` dt

~ t221!m1 1/2eixt. ~75!

The result satisfies the Bessel equation for Imxt→` for utu→`.A similar calculation forHm

(2) gives

Hm(2)~x!52

1

2p i

2

ApS 2

xD m

eipmGS m11

2D E(`,11,`)

dt

~ t221!m1 1/2e2 ixt ~76!

or, for Rem,12,

Hm(1)~x!5

2i

ApS 2

xD m 1

GS 1

22m D E1

` dt

~ t221!m1 1/2e2 ixt. ~77!

For x real and2 12,Rem,1

2, ~75! and~77! can be combined to obtain the representations forJm andYm noted in Ref. 6, 6.13~3! and ~4!,

Jm~x!52

ApS 2

xD m 1

G~ 12 2m!

E1

`

dtsinxt

~ t221!m1 1/2, ~78!



138.38.171.231 On: Wed, 26 Nov 2014 13:41:43

Ym~x!522

ApS 2

xD m 1

G~ 12 2m!

E1

`

dtcosxt

~ t221!m1 1/2. ~79!

A different set of integral representations can be obtained by considering the action of theinverse operatorP1

2m21/2 on t1/2H1/2(1,2)(x),

P12m2 (1/2) t1/2H1/2

(1,2)~x!5t2mH2m(1,2)~x!. ~80!

Using

H2m(1) ~x!5eipmHm

(1)~x!, H2m(2) ~x!5e2 ipmHm

(2)~x! ~81!

and following the above-presented manipulations, we obtain the integral representations

Hm(1)~x!5

i

p

2

ApS x

2D m

e22p imGS 1

22m D E

(`,11,`)dt ~ t221!m2 (1/2)eixt, ~82!

Hm(2)~x!5

i

p

2

ApS x

2D m

GS 1

22m D E

(`,11,`)dt ~ t221!m2 (1/2)e2 ixt, ~83!

where, for convergence,t must approach on the contours in~82! and~83! with Im xt→1` andIm xt→2`, respectively. These expressions are equivalent to the representations 6.11~4! and6.11~5! in Ref. 6 obtained from Hankel’s representation for the Bessel functions.@Watson uses adifferent phase convention in his 6.11~4! which is equivalent to replacingt21 in ~82! by e2p i(t21). Watson’s 6.11~5! is obtained from~83! by the substitutionst21→eip(u11) and t11→e2 ip(u21).] Other results, for example, Scha¨fli’s integral for Km @Ref. 6, 6.15~4!#, can beobtained from these. See Watson.6

D. Riemann-type relations for Bessel functions

1. Relations for P Ám

For Riemann-type fractional operators, the roles ofP1m andP2

m are essentially reversed, andthe relations apply to different Bessel functions. The action of the Riemann operatorP2

m is givenby ~16! or ~17! and ~45!. We will use the expression in~17! which gives

P2m tnZn~x!5tn2mZn2m~x!5

1

2p ieipmG~m11!S t

xD n

3ECR

du

um11 S x222ux

t D n/2

ZnSAx222ux

t D , ~84!

whereCR5(u(x,t),01,u(x,t)). The end pointsu(x,t) of the integration must be chosen suchthat tm2n times the integral gives a solutionZn2m(x) of the Bessel equation. This requires that

S x222x

tuD n11 d

du F S x222x

tuD 2n/2

ZnSAx222x

tu D G50 ~85!

at the end points of the integration contour.@The integrand vanishes foru5xt/2, suggesting thatvalue for u(x,t). With that assumed, the precise condition for a solution of Bessel’s equationfollows by scaling the integration variable as in~86! to eliminatex and t from the limits ofintegration, applying the relevant operator, and then undoing the scaling in the resulting condi-



138.38.171.231 On: Wed, 26 Nov 2014 13:41:43

tion.# This condition can be satisfied for the Bessel functionsZn5Jn , I n for end pointsu(x,t)5xt/2 in CR5 provided Ren.21. The condition cannot be satisfied for other choices of theBessel functionZn .

We can easily show that the Riemann operator defined by~84! satisfies the product ruleP2

l P2m 5P2

l1m provided we choose the end points in the integrations properly. It is convenient inthis to assume that Rel,0 and Ren,0, conditions which can be attained using~16!. The contourintegrals can then be converted into ordinary integrals. The action of the group operatore2vP2 onthe integrand in~84! changesx2 to x222vx/t, but does not affectx/t sinceP2(x/t)s50. As aresult, the parametersu andv appear only in the sumu1v as required by the operator relatione2uP2e2vP25e2(u1v)P2. It is then straightforward to show that the double integral can bereduced to the product of a beta function and an integral of the form in~84!, and gives a solutionof the Bessel equation equal totn2l2mZn2l2m(z), provided the end points in the successiveintegrations are taken asv0(x,t,u)5xt/22u, u0(x,t)5xt/2. The sublety is that the end point ofthe first integration depends on the variable in the second. The result gives an example of theformal relation in~18! which generalizes the product rule for Riemann fractional integrals.

A change of the integration variable tov52u/xt converts~84! to the simpler form

Zn2m~x!51

2p ieipmG~m11!S 2

xD mE(1,01,1)

dvvm11 ~12v !n/2Zn~xA12v !. ~86!

Alternatively, for Rem,0, we can collapse the integration contour in~84! and change tovariablesz5x2, v25x222ux/t to put the result in the form of a standard Riemann fractionalintegral,

z(n2m)/2Zn2m~Az!52m

G~2m!E

0

z dv~z2v !m11 vn/2Zn~Av !, ~87!

Rem,0, Ren.21. This reproduces Ref. 5, 13.1~63! and 13.1~83! for Zn5Jn andZn5I n whenmis replaced by2m in accord with the convention used there.

For Zn5Yn , Kn , the action of the Bessel operator in the variablex5Az on the function‘‘ Zn2m’’ defined by ~86! leaves a term proportional tox2n2m. The result is an inhomogeneousBessel equation with a solution which involves a sum of a functionJn2m or I n2m and the Lommelfunctionss2n2m11,n2m(x) ~Ref. 6, Sec. 10.7!. The fractional integral Ref. 5, 13.1~73! is of thistype.

An analysis similar to that above shows that the integral in~65!, taken on a Riemann-typecontour with end points atu52x/2t satisfies an inhomogeneous rather than homogeneous Besselequation of ordern1m. The general solution involves Lommel functions, and there is no Rie-mann definition forP1

m acting on Bessel functions alone.

2. Riemann-type integral representations for Bessel functions

The operator relationP2m tnZn(x)5tn2mZn2m(x) immediately gives integral representations

for Jn and I n . We will start with the functions of ordern52 12,

J21/2~x!52

Ap

cosx

x1/2 , I 21/2~x!52

Ap

coshx

x1/2 . ~88!

Choosingm52l2 12, ~86! then gives

Jl~x!51

2p ie2 ip(l1 1/2)GS 2l1

1

2D 2

ApS x

2D lE(1,01,1)

dv vl2 (1/2)~12v !2 1/2cos~xA12v !

~89!

for generall, or, replacingv by 12t2,



138.38.171.231 On: Wed, 26 Nov 2014 13:41:43

Jl~x!5G~ 1

2 2l!

ipG~ 12!

E(0,11,0)

dt ~12t2!l2 (1/2) cosxt, ~90!

52

ApGS l11

2D S x

2D lE0

1

dt ~12t2!l2 (1/2) cosxt ~91!

for Rel.212. The first is a standard Poisson-type integral representation forJl(x) @Ref. 6, 3.3~2!#.

The second gives the generalization of Ref. 6, 6.1~6!.Similarly, from P2

2l2(1/2)t21/2I 21/2(x)5tlI l(x),

I l~x!51

2p ie2 ip(l1 1/2)GS 2l1

1

2D3S x

2D lE(1,01,1)

dv vl2 (1/2)~12v !2 1/2cosh~xA12v !, ~92!

5G~ 1

2 2l!

ipG~ 12!

E(0,11,0)

dt ~12t2!l2 (1/2) coshxt ~93!

52

ApGS l11

2D S x

2D lE0

1

dt ~12t2!l2 (1/2) coshxt, Rel.21

2.

~94!

IV. SUMMARY

Many of the properties of the special functions arise from their connection to Lie groups.1–3

Their differential recurrence relations, for example, reflect the action of particular multivariableoperatorsD in the associated Lie algebra, the so-called stepping operators, on the functions in therelevant class. We have given general definitions of fractional operatorsDl in the context of Lietheory, and explored their formal properties. Our Weyl- and Riemann-type fractional operatorsgeneralize the single-variable Weyl and Riemann fractional integralsW2l andR2l ~Ref. 5, Chap.13!. The operatorsDl change the indices on the special functions by fractional displacementsrelated tol, and provide useful connections between functions in different realizations of the Liealgebra.

We have illustrated the usefulness of the fractional operators in the case of the Euclideangroup E~2! and the Bessel functions, and find that they contribute to a coherent overall picture ofmany relations among the Bessel functions as interpreted in the group context. For example, theformal relationsP6

l tmZm(x)5tm6lZm6l(x) give the integral relations connecting Bessel func-tions of different orders. When reduced to the single variablex, these generalize known fractionalintegral relations. Used with simple choices ofm andl, with Zm an elementary function, they leadimmediately to the standard integral representations for the various Bessel functions, representa-tions which are derived in Ref. 6 from quite different starting points using different methods. Inaddition, the action of the elementse2uP6 on the functionstmZm(x) gives generating functions forthe Bessel functions~the Lommel expansions!, while the Bessel equation itself is the statementthat the Casimir operatorP1P2 and the rotation operatoriJ3 have fixed values21 andm.

The applications of the fractional group operators will be extended in a following paper to theassociated Legendre functions in the somewhat more complicated case of SO~2,1! and its confor-mal extension.



138.38.171.231 On: Wed, 26 Nov 2014 13:41:43

ACKNOWLEDGMENTS

This work was supported in part by the U. S. Department of Energy under Grant No. DE-FG02-95ER40896, and in part by the University of Wisconsin Graduate School with funds grantedby the Wisconsin Alumni Reseach Foundation. The author would like to thank the faculty of theInstitute for Advanced Study for their hospitality during the fall term of 1975 when the initialstages of this work were carried out, and the Aspen Center for Physics for its hospitality whileparts of the final work were done.

1N. J. Vilenkin, Special Functions and the Theory of Group Representations~American Mathematical Society, Provi-dence, 1968!.

2W. Miller, Jr., Lie Theory and Special Functions~Academic, New York, 1968!.3J. D. Talman,Special Functions. A Group Theoretical Approach~Benjamin, New York, 1968!.4W. Miller, Jr., Symmetry and Separation of Variables~Addison–Wesley, Reading, MA, 1977!.5Tables of Integral Transforms, edited by A. Erde´lyi ~McGraw–Hill, New York, 1953!.6G. N. Watson,Theory of Bessel Functions~Cambridge University Press, Cambridge, 1966!.7R. Askey,Othogonal Polynomials and Special Functions~Society for Industrial and Applied Mathematics, Philadelphia,1975!.

8R. Gilmore,Lie Groups, Lie Algebras, and Some of Their Applications~Wiley, New York, 1974!.9Higher Trancendental Functions, edited by A. Erde´lyi ~McGraw–Hill, New York, 1953!.

10L. Weisner, Can. J. Math.11, 148 ~1959!.11Handbook of Mathematical Functions, edited by M. Abramowitz and I. A. Stegun~Dover, New York, 1967!.



138.38.171.231 On: Wed, 26 Nov 2014 13:41:43

Documents

Fractional operators and special functions. I. Bessel functions