derivadas de cualquier orden 1f1

Derivatives of any order of the confluent hypergeometric function1F1(a,b,z) with respect to the parameter a or bL. U. Ancarani and G. Gasaneo

Citation: J. Math. Phys. 49, 063508 (2008); doi: 10.1063/1.2939395 View online: http://dx.doi.org/10.1063/1.2939395 View Table of Contents: http://jmp.aip.org/resource/1/JMAPAQ/v49/i6 Published by the American Institute of Physics.

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Derivatives of any order of the confluent hypergeometricfunction 1F1a ,b ,z with respect to the parametera or b

L. U. Ancarani1 and G. Gasaneo2,a1Laboratoire de Physique Molculaire et des Collisions, Universit Paul Verlaine-Metz,57078 Metz, France2Departamento de Fsica, Universidad Nacional del Sur and Consejo Nacionalde Investigaciones Cientficas y Tcnicas, 8000 Baha Blanca, Buenos Aires, ArgentinaReceived 19 March 2008; accepted 9 May 2008; published online 20 June 2008

The derivatives to any order of the confluent hypergeometric Kummer functionF= 1F1a ,b ,z with respect to the parameter a or b are investigated and expressedin terms of generalizations of multivariable Kamp de Friet functions. Variousproperties reduction formulas, recurrence relations, particular cases, and series andintegral representations of the defined hypergeometric functions are given. Finally,an application to the two-body Coulomb problem is presented: the derivatives of Fwith respect to a are used to write the scattering wave function as a power series ofthe Sommerfeld parameter. 2008 American Institute of Physics.DOI: 10.1063/1.2939395

I. INTRODUCTION

The confluent hypergeometric function F= 1F1a ,b ;z or alternatively Whittakers functionhas been studied in great detail from its mathematical point of view.13 The Kummer function, asit is also named, is closely related to a fundamental problem of quantum mechanics: the two-bodyCoulomb problem. It is well known that the closed form solution of the two-body CoulombSchrdinger equation both in spherical and parabolic coordinates are written in terms of theKummer function F.4 To perform different types of physical studies see, e.g., Refs. 59 andapplications,1013 it is necessary to know the mathematical properties of the function. Usually F isconsidered as a function of variable z; however, in some physical applications, the first a orsecond b parameter may be the physical variable, as, for example, when Coulomb solutions areextended to the complex plane of the energy5,7 or the angular momentum.5

The first and the nth derivatives with respect to the variable z is known in a compact forme.g., Ref. 1. The derivatives with respect to the first or second parameter, on the other hand, havebeen less studied simply because the mathematical formulation is more difficult see below.These derivatives, however, are needed in different physical applications, the Coulomb Born seriesbeing probably the most well known example. Several forms of the first derivatives have beengiven in the literature; none of them, however, is compact except possibly in some special cases.In this contribution we address this issue by providing a compact form not only for the first butalso for the nth derivatives. The usefulness of the investigation is illustrated by considering thetwo-body Coulomb wave function as a function of the Sommerfeld parameter.

We shall use the following notation for the nth derivatives:

Gn = Gna,b;z =dn1F1ab ;z

dan, 1

aElectronic mail: [email protected].

JOURNAL OF MATHEMATICAL PHYSICS 49, 063508 2008

49, 063508-10022-2488/2008/496/063508/16/$23.00 2008 American Institute of Physics


Hn = Hna,b;z =dn1F1ab ;z

dbn. 2

Let us recall the definition of the confluent hypergeometric function as a power series on thevariable z:

F = 1F1ab ;z = n=0

ann!bn

zn, 3

where an=a+n /a is the Pochhammer symbol defined in terms of the gamma function.1A first approach to get the first derivative G1, discussed, for example, in Ref. 1, is based on

the use of the derivative of the Pochhammer symbol:

danda

= ana + n a , 4

and its definition in terms of the digamma function z.14 With the previous definition, the firstderivatives of F with respect to a or b are

G1 = n=0

anbn

a + n azn

n!=

n=0

anbn

a + nzn

n!a1F1ab ;z , 5a

H1 = n=0

anbn

b + n bzn

n!=

n=0

anbn

b + nzn

n!+b1F1ab ;z . 5b

We have therefore an infinite series containing the digamma function.14 An alternative form isobtained by using the recurrence formula 6.3.6 of Ref. 14:

G1 = n=0

anbn

zn

n!p=0n1 1

p + a, 6a

H1 = n=0

anbn

zn

n!p=0n1 1

p + b. 6b

In one form or the other, the generalization to the nth derivative is particularly cumbersome.A second approach makes use of Whittaker functions:

M,z = ez/2z1/2+1F1 12 + 1 + 2 ;z . 7In the book The Confluent Hypergeometric Function,3 Buchholz studied briefly the first derivativeof the Whittaker function with respect to and gave expression in terms of the digamma function.This author also related the first derivative with respect to and with inhomogeneous differ-ential equations, but no explicit and compact expressions were given for the derivatives G1 andH1. Moreover, the generalization to the nth derivative was not provided.

An algorithm for the computation of Gn was given recently by Abad and Sesma.15 Theymade use of a convergent expansion of Whittakers function in series of Bessel functions and ofthe properties of Buchholz polynomials pj

2z j=0,1 ,2 , . . . . The main result is

063508-2 L. U. Ancarani and G. Gasaneo J. Math. Phys. 49, 063508 2008


M,n z =

dnM,zdn

= 1nz+1/2+nj=0

pj2z

2 j2 + 1 j+n 0F1 2 + 1 + j + n ; z . 8

The algorithm is based on this expansion, truncated at a conveniently large value jmax.In this paper explicit expressions for the nth derivatives Gn and Hn are given in Secs. II and

III. The connection with Kamp de Ferit-like multivariable hypergeometric functions is given inSec. IV, where different properties of these functions are presented. As application of these results,the scattering wave function for the two-body Coulomb problem is expressed as a power series ofthe Sommerfeld parameter up to order 2 Sec. V. A summary of the results is given in Sec. VI.

II. FIRST DERIVATIVE WITH RESPECT TO a OR b

With an approach based on the solution of inhomogeneous differential equations related to theconfluent hypergeometric function F, we give here a compact form for the first derivatives G1and H1. The procedure will then be used in the next section to find expressions for the higherderivatives Gn and Hn.

We start from the differential equation satisfied by the confluent hypergeometric function F;

z d2dz2 + b z ddz aF = 0. 9Since F is an analytic function of z and a,1 taking the derivative of Eq. 9 with respect to the firstparameter a, one finds

z d2dz2 + b z ddz aG1 = F = m1=0 am1

bm1

zm1

m1!10

the use of m1 as index will become clear in the next section. Similarly, taking the derivative ofthe differential equation 9 with respect to b F has poles and is not defined for b=0,1,2, . . . Ref. 1, we have

z d2dz2 + b z ddz aH1 = dFdz = ab 1F1a + 1b + 1 ;z = ab m1=0 a + 1m1

b + 1m1

zm1

m1!. 11

Now, according to Eq. 4.162 of Ref. 16, the solution of the inhomogeneous differential equation

z d2dz2 + b z ddz ay = zm1 12is given by Eq. 4.163:

y = m1+1a,b;z =zm1+1

1 + m1b + m1 2F2 1,a + 1 + m12 + m1,b + 1 + m1 ;z . 13

Since the differential equation 10 is linear, the solution for G1 reads

G1 = m1=0

am1m1!bm1

m1+1a,b;z , 14

that is to say,

G1 =z

b1

m1=0

am11m1b + 1m12m1

zm1

m1! 2F2 1,a + 1 + m12 + m1,b + 1 + m1 ;z . 15

Similarly, for H1 we find

063508-3 Derivatives of the Kummer function J. Math. Phys. 49, 063508 2008


H1 = a

bz

b1

m1=0

a + 1m1bm11m1b + 1m1b + 1m12m1

zm1

m1! 2F2 1,a + 1 + m12 + m1,b + 1 + m1 ;z . 16

These expressions, to the best of our knowledge, have not been given before. Expanding the 2F2hypergeometric function in series index m2 and after some algebraic manipulations, one finds

G1 =z

b1

m1=0

m2=0

1m11m2am1a + 1m1+m2a + 1m12m1+m2b + 1m1+m2

zm1+m2

m1!m2!, 17a

H1 = 1b

a

bz

m1=0

m2=0

1m11m2bm1a + 1m1+m2b + 1m12m1+m2b + 1m1+m2

zm1+m2

m1!m2!. 17b

These double series could have been obtained also starting directly from expressions 6a and 6busing the property

n=0

k=0

n

Bk,n = n=0

k=0

Bk,n + k . 18

However, as mentioned in the Introduction, this technique does not allow for an easy generaliza-tion to the nth derivatives. Moreover, these double series can be related to the following hyper-geometric function in two variables:

1a1,a2b1,b2c1d1,d2

;x1,x2 = m1=0

m2=0

a1m1a2m2b1m1b2m1+m2c1m1d1m1+m2d2m1+m2

x1m1x2

m2

m1!m2!, 19

which, as we shall see in Sec. IV, is a Kamp de Friet-like function. Hence, in terms of 1, thefirst derivatives G1 and H1 read

G1 =z

b11 1,1a,a + 1

a + 12,b + 1 ;z,z , 20aH1 =

a

bz

b11 1,1b,a + 1

b + 12,b + 1 ;z,z . 20bAn alternative formulation of the solution to the inhomogeneous differential equation Eq. 12

is given by a finite sum of powers of z see Eq. 4.172 of Ref. 16:

m1+1a,b;z = 1a

m1!bm1a + 1m1

n=0

m1 anbn

zn

n!=

1a

m1!bm1a + 1m1

Fm1a,b,z . 21

As showed notin Ref. 16 Eq. 4.175, the functions m1+1a ,b ;z and m1+1a ,b ;z are linked toeach other through

m1+1a,b;z = m1+1a,b;z 1a

m1!bm1a + 1m1

1F1a,b;z . 22

The function m1+1a ,b ;z can be considered as an asymptotic solution of Eq. 12 as it is derivedfrom a series in inverse powers of z; note that limm Fma ,b ,z= 1F1a ,b ;z. Bothm1+1a ,b ;z and m1+1a ,b ;z are closely related to the inhomogeneous Whittaker functions Sand R defined by Buchholz.3 In terms of the finite sum Fm1, the derivative G1 given by Eq. 14can thus be expressed as



G1 = 1a

m1=0

am1a + 1m1

Fm1a,b,z . 23

After some algebraic calculations and the application of series rearrangement techniques,17 theG1 function can be reduced to

G1 = 1a

m1=0

am1bm1

3F2 1,a,a + m1a + 1,a + 1 + m1

;1 zm1m1!

. 24

For the particular argument of z=1, the function the 3F2 simplifies18

and an expression in terms ofdigamma functions, leading to Eq. 5a, can be obtained. Following the same procedure, a similarrepresentation for H1 can be obtained.

III. nTH DERIVATIVE WITH RESPECT TO a OR b

Without much effort, one may generalize the result to the nth derivative with respect to a orb. Following the same procedure, we define first the inhomogeneous differential equation satisfiedby each derivative order.

Consider the second derivative G2. Differentiating with respect to a Eq. 10 satisfied byG1, we have

z d2dz2 + b z ddz aG2 = G1 + dG0da = 2G1, 25where we used F=G0. Taking again the derivative with respect to a, we obtain for the thirdderivative

z d2dz2 + b z ddz aG3 = G2 + 2dG1da = 3G2, 26and, following the same procedure, we find that the nth derivative satisfies the general differentialequation

z d2dz2 + b z ddz aGn = nGn1. 27Proceeding as for G1, in Appendix A we give the detailed calculations that lead to the

following explicit expressions for G2:

G2 =z2

b2

m1=0

m2=0

m3=0

1m11m21m33m1+m2+m3b + 2m1+m2+m3

am1a + 1m1+m2a + 2m1+m2+m3

a + 1m1a + 2m1+m2

zm1+m2+m3

m1!m2!m3!. 28

It can be easily verified by induction that the general expression for Gn n1 reads

Gn =zn

bn

m1=0

mn+1=0

1m11m2 1mn+1n + 1m1+m2++mn+1b + nm1+m2++mn+1

am1a + 1m1+m2 a + nm1+m2++mn+1

a + 1m1 a + nm1++mnzm1+m2++mn+1

m1!m2! mn+1! . 29

Following a similar procedure, one finds the set of equations satisfied by Hn:



z d2dz2 + b z ddz aH1 = dFdz , 30az d2dz2 + b z ddz aH2 = 2dH1dz , 30b

,

z d2dz2 + b z ddz aHn = ndHn1dz . 30cThe solution for 30a was given above by Eq. 20b. The solution for 30b reads

H2 = 122b2

a

bz

m1=0

m2=0

m3=0

1m11m21m32m1+m2+m3b + 1m1+m2+m3

bm1bm1+m2a + 1m1+m2+m3

b + 1m1b + 1m1+m2

zm1+m2+m3

m1!m2!m3!, 31

and, by induction, the solution for the nth derivative Hn is

Hn = 1nn!bn

a

bz

m1=0

mn+1=0

1m11m2 1mn+12m1+m2++mn+1b + 1m1+m2++mn+1

bm1bm1+m2 bm1+m2++mna + 1m1+m2++mn+1

b + 1m1b + 1m1+m2 b + 1m1+m2++mnzm1+m2++mn+1

m1!m2! mn+1! . 32

It is worth mentioning that the system of equations for the nth derivatives Eqs. 10 and2527 for Gn and Eqs. 30a30c for Hn are all of order 2 and could be numerically solvedwith well known methods.

IV. CONNECTION WITH MULTIVARIABLE HYPERGEOMETRIC FUNCTIONS ANDVARIOUS PROPERTIES

It is interesting to notice that the expressions given above for the derivatives of F with respectto a and b can be expressed in terms of generalizations of multivariable Kamp de Friet hyper-geometric functions. In Sec. II we have written the first derivatives G1 and H1 in terms of thetwo-variable hypergeometric 1. Similarly, for each nth derivative n2, we may associate adifferent multivariable hypergeometric function which we shall name n. Several properties ofthese functions will be presented in this section.

A. The definition of Gn and Hn in terms of hypergeometric functionsThe function 1, defined by Eq. 19, results from the application of the rule used by

Appell19 to the product of the generalized confluent hypergeometric functions 3F3 and 2F2,

3F3a1,a2,a3c1,c2,c3

;x1 2F2b1,b2d1,d2 ;x2 = m1=0

m2=0

x1

m1x2m2

m1!m2!a1m1a2m1a3m1b1m2b2m2c1m1c2m1c3m1d1m2d2m2

.

Using the replacements

a3m1b1m2 a3m1+m2,



c3m1d2m2 c3m1+m2,

c2m1d1m2 c2m1+m2,the following coefficient is generated:

a1m1a2m1a3m1b1m2b2m2c1m1c2m1c3m1d1m2d2m2

a1m1a2m1b2m2a3m1+m2

c1m1c2m1+m2c3m1+m2.

According to the theory presented in Ref. 17, the fact that the functions we started from areconfluent hypergeometric functions ensures that the function 1 given by Eq. 19 is also aconfluent hypergeometric function whose convergency radius is infinity in x1 and x2. The function1 is a Kamp de Friet function in two variables.

Similarly, for the second order derivatives G2 and H2, we may introduce the function

2a1,a2,a3b1,b2,b3c1,c2d1,d2

;x1,x2,x3=

m1=0

m2=0

m3=0

a1m1a2m2a3m3b1m1b2m1+m2b3m1+m2+m3c1m1c2m1+m2d1m1+m2+m3d2m1+m2+m3

x1m1x2

m2x3m3

m1!m2!m3!, 33

which can be generated using Appells technique to the product of generalized confluent hyper-geometric functions 4F4, 3F3, and 2F2. Generalizing to the nth derivative, we introduce the func-tion n defined by

na1,a2, . . . ,an+1b1, . . ,bn+1c1, . . . ,cnd1,d2

;x1,x2, . . ,xn+1=

m1=0

mn+1=0

x1

m1x2m2 xn+1mn+1

m1!m2! mn+1!

a1m1a2m2 an+1mn+1b1m1b2m1+m2 bn+1m1+m2++mn+1

c1m1c2m1+m2 cnm1+m2++mnd1m1+m2++mn+1d2m1+m2++mn+1, 34

which results from the application of Appells technique to the product of n+2Fn+2,. . ., 3F3, and 2F2.The way in which the coefficients are combined is easily induced from the series given here.

In terms of these generalized Kamp de Friet-like hypergeometric functions, the derivativesGn given in the previous section read

G1a,b;z =z

b11 1,1a,a + 1

a + 12,b + 1 ;z,z , 35aG2a,b;z =

z2

b221,1,1a,a + 1,a + 2

a + 1,a + 23,b + 2 ;z,z,z , 35bGna,b;z =

zn

bnn 1,1, . . . ,1a,a + 1, . . . ,a + n

a + 1,a + 2, . . . ,a + nn + 1,b + n ;z,z, . . . ,z . 35cSimilar expressions can be obtained for Hn following the same procedure:

Hna,b;z = 1nn!bn

a

bzn 1,1, . . . ,1b,b, . . . ,b,a + 1

b + 1,b + 1, . . . ,b + 12,b + 1 ;z,z, . . . ,z . 36



With these derivatives one may thus provide Taylor expansions for the function F in powerseries of a around a0 or of b around b0,

F = n=0

a a0n

n!Gna0,b;z , 37a

F = n=0

b b0n

n!Hna,b0;z . 37b

Here we are using the notation

G0a0,b;z = 1F1a0b ;z and H0a,b0;z = 1F1 ab0 ;z .In Sec V, we shall use expansion 37a for a0=0.

B. Particular value: a=0

The hypergeometric functions n defined by Eq. 34 depend on a large number of param-eters. However, in the expression of the derivatives Gn and Hn Eqs. 35c36, only theparameters a and b of the initial function F are actually variable. Since in the calculations to beperformed in Sec. V we will need the evaluation of Gna ,b ;z with n1 for a=0, we nowprovide explicit formulas for this situation.

The reduction formulas for the nth derivatives Gn in the case a=0 can be related to those,presented in Appendix B, corresponding to the n functions. The results are

G10,b;z =z

b11 1,10,1

12,b + 1 ;z,z = zb1 2F2 1,12,b + 1 ;z , 38aG20,b;z =

z2

b21 1,11,2

23,b + 2 ;z,z , 38bGn0,b;z =

zn

bnn1 1, . . . ,11, . . . ,n

2,3, . . . ,nn + 1,b + n ;z, . . . ,z . 38cThe nth derivatives Gn n1 for a=0, defined in terms of n, are thus expressed in terms ofa function n1.

C. Recurrence relations for Gn

1. Relations for general a and bStarting from the recurrence relations for the confluent hypergeometric function F, recurrence

relations for the 1 function can be easily deduced. For example, consider the contiguous rela-tion 13.4.4 of Ref. 14:

b1F1ab ;z b1F1a 1b ;z z1F1 ab + 1 ;z = 0. 39If we take the derivative with respect to a, we find

bG1a,b;z bG1a 1,b;z zG1a,b + 1;z = 0, 40

and hence, using Eq. 20a, a relation for 1:



1 1,1a,a + 1a + 12,b + 1 ;z,z 11,1a 1,aa2,b + 1 ;z,z zb + 11 1,1a,a + 1a + 12,b + 2 ;z,z = 0.

41

Similarly using relation 13.4.13 of Ref. 14, one finds a relation providing the derivative of the1 function with respect to z,

z

bddz1 1,1a,a + 1

a + 12,b + 1 ;z,z =11,1a,a + 1a + 12,b ;z,z 21 1,1a,a + 1a + 12,b + 1 ;z,z .42

Several other relations can be obtained in a similar way and easily generalized to n.

2. Relations for a=0Let us introduce also the following function:

Gn,ma,b;z =zn

bnn 1,1, . . . ,1a,a + 1, . . . ,a + n

a + 1,a + 2, . . . ,a + nm + 1,b + n ;z,z, . . . ,z . 43By inspection of relation 38c, we see that

Gn,m0,b;z =1b

zGn1,m1,b + 1;z . 44

As a consequence, we have the following recurrence relation for n2:

Gn0,b;z = Gn,n0,b;z =1b

zGn1,n1,b + 1;z . 45

D. Series and integral representations in terms of one-variable hypergeometricfunctions

In this section we provide series and integral representations for 1 in terms of well knownone-variable hypergeometric functions generalization to n can be easily obtained. Using seriesrearrangement techniques and different properties for the Pochhammer symbols, it is easy to verifythat the following series representations hold for the 1 function:

1a1,a2b1,b2c1d1,d2

;x1,x2 = m1=0

a1m1b1m1b2m1c1m1d1m1d2m1

x1m1

m1! 2F2 a2,b2 + m1d1 + m1,d2 + m1 ;x2

46a

= m2=0

a2m2b2m2d1m2d2m2

x2m2

m2! 3F3 a1,b1,b2 + m2

c1,d1 + m2,d2 + m2 ;x1 . 46b

Definitions for G1 and H1 result immediately from Eqs. 20a and 20b. Equation 46a is theformulation encountered in Eqs. 15 and 16. The rate of convergence of these series is fasterthan the one corresponding to the double series of Eq. 19 as illustrated in Fig. 1, where we plot1 given in Eq. 35a as a function of x1=x2= ix for fixed values of the parameters a= i /0.75 andb=1 the value b=1 and the purely imaginary values of a and argument correspond to theconfluent hypergeometric function appearing in the Coulomb problem, see Sec. V. We used m1=m2 up to 20 to evaluate 1 with the double series of Eq. 19 and m1 up to 20 or 60 with Eq.46a. As can be noticed from the figure, a much smaller number of terms are needed when the



single series is used. This is a natural consequence of the fact that the whole series in m2 has beensummed up.

A numerically useful integral representation for 1 can be obtained starting from Eq. 46a.Combining the integral representations for the 2F2 see Ref. 20, p. 854:

2F2,a,b ;x2 = 01

dt1 t1t1 1F1ab ;x2t , 47where R0, R0, and the one corresponding to the 1F1 Ref. 1:

1F1ab ;z = bab a01

du1 uba1ua1ezu, 48

where Rba0, Ra0, we obtain

2F2,a,b ;z = b ab a01

dt1 t1t10

1

du1 uba1ua1eztu.

49

The condition over the parameters can be removed by performing the integrations over contourson the complex plane.1 With this representation, the series of Eq. 46a becomes

FIG. 1. Color online The 1 function given in Eq. 35a is plotted vs the argument x1=x2= ix for fixed values of theparameters a= i /0.75 and b=1: it is is calculated with the double series, Eq. 19 with m1=m2 up to 20 solid line andsquares and the series of Eq. 46a with m1 up to 20 solid lines and circles or 60 solid lines and triangles.



1a1,a2b1,b2c1d1,d2

;x1,x2=

m1=0

a1m1b1m1b2m1c1m1d1m1d2m1

x1m1

m1!d1 + m1d2 + m1

a2d1 + m1 a2b2 + m1d2 b2

0

1

dt1 td1+m1a21ta210

1

du1 ud2b21ub2+m11ex2tu, 50

which after algebraic manipulations converts into

1a1,a2b1,b2c1d1,d2

;x1,x2=

d1d2a2d2 b2b2d1 a2

0

1

dt1 td1a21ta210

1

du1 ud2b21ub21

ex2tu2F2 a1,b1c1,d1 a2

;x11 tu . 51This integral representation is particularly useful to numerically evaluate 1, and thus G1 andH1 through the use of Eqs. 20a and 20b. Following similar procedures, integral and seriesrepresentations of this type can be obtained for n, and hence for Gn and Hn.

E. Special cases

When b=a, the function F reduces to an exponential F=ez, so that there is no parameterdependence and G1=H1=0. Although the result is trivial, it should appear also through the 1.Indeed, we have

dda 1

F1aa ;z = G1a,a;z + H1a,a;z , 52

and using Eqs. 20a and 20b, we immediately find 0.When b=a1, the function F reduces to

1F1 aa 1 ;z = ez1F1 1a 1 ; z = ez1 + za 1 . 53

The derivative with respect to a is G1=zez / a12, a result which can be easily found byapplying, for example, Eq. 20b.

Another interesting situation is when b=a+1 since the F function is related to the incompletegamma function

1F1 aa + 1 ;z = a zaa, z . 54

In this case the derivative of the confluent hypergeometric function with respect to a reads

dda 1

F1 aa + 1 ;z = G1a,a + 1;z + H1a,a + 1;z = za + 121 1,1a,a + 1a + 22,a + 2 ;z,z .

55

Hence, the first derivative with respect to a of the incomplete Gamma function reads



ddaa, z = a, zln z 1

a

+ za

a

z

a + 121 1,1a,a + 1

a + 22,a + 2 ;z,zand the nth derivative can be easily derived with the results given in the previous sections.

Other special cases can be considered in a similar manner.

V. APPLICATION TO THE BORN-LIKE SERIES FOR THE TWO-BODY COULOMBSCATTERING WAVE FUNCTION

As we mentioned in the Introduction, a close connection exists between the Kummer functionand the two-body Coulomb problem. Let r and k respectively represent the relative vector positionand momentum between an electron and a heavy nucleus of charge Z placed at the origin ofcoordinates. The solution for the scattering problem in parabolic coordinates is given by4

+,k,r = Neikr1F11 ;ik , 56where outgoing wave boundary conditions are considered. The parabolic coordinates are =r+k r, =rk r, and tan =y /x, where the Cartesian coordinates x and y correspond to theposition of the particle relative to the reference center.4 Here atomic units =me=e=1 are used,so that the electron charge is equal to 1; = iZ /k is the Sommerfeld parameter we haveincluded i in its definition for convenience. The normalization factor N is defined in terms ofthe gamma function as

N = ei/21 . 57

Let us consider the expansion of the scattering wave function + ,k ,r in power series ofthe Sommerfeld parameter :

+,k,r = eikr0+k,r +1+k,r +2+k,r22 + , 58where l+k ,r is given by

l+k,r = dl+,k,rdl =0. 59Such an expansion is interesting as it is related to the Born series for the Coulomb problem. Inorder to get analytical expressions for the different orders l+k ,r of 58, we shall need the nthderivatives Gn with respect to the first parameter of the confluent hypergeometric function Fappearing in 56. Indeed, they appear in the Taylor expansion 37a of F around a0=0; we have

F = G00,1;ik + G10,b;ik +2

2G20,1;ik + , 60

where G0= 1F10,1 ; ik=1. We also need to expand N in power series of :

N = N0 + N1 + N22

2+ , 61

where

N0 = 1, 62a

N1 = i2 , 62b



N2 = 2 i 212 , 62cand represents the Euler gamma constant.14

Combining the two series expansions 60 and 61 and comparing with 58, the followingexpressions are readily deduced for the first three terms:

0+k,r = N0G00,1;ik , 63a

1+k,r = N1G00,1;ik + N0G10,1;ik , 63b

2+k,r = N2G00,1;ik + 2N1G10,1;ik + N0G20,1;ik . 63c

Using 62a62c and the reduction formulas 38a and 38b for, respectively, G1 and G2, wefind

0+k,r = 1, 64a

1+k,r = i2 + ik2F21,12,2 ;ik , 64b2+k,r = 2 i 212 + 2 i2 ik2F21,12,2 ;ik

+ik2

211,11,223,3 ;ik,ik . 64c

The numerical evaluation of 1 can be easily performed using one of the representations givenin Sec. IV D. The function 2F2 appearing in 1+ and 2+ can be reduced to simpler functions:

21

2F21,12,2 ;ik = log ik 0, ikik ,where a ,z represents the incomplete gamma function.14

Explicit expressions of n+k ,r have been given here up to order n=2; higher orders can beeasily obtained in terms of the generalized hypergeometric functions n or of n1 if thereduction formula 45 is used. In order to reduce the difficulties in their evaluations, furtherinvestigation of the properties of the multivariable hypergeometric functions n defined here isnecessary.

VI. SUMMARY

We have given closed form formulas for the nth derivatives of the Kummer function withrespect to its parameters a and b. These derivatives are expressed in terms of multivariable Kampde Friet-like hypergeometric functions, named here n. For 1, which is related to the firstderivatives, various types of properties such as recurrence relations and series and integral repre-sentations are provided, and some special cases are discussed. The system of ordinary differentialequations satisfied for the derivatives of order n is also given.

The above mathematical study is applied to the physical case of the two-body Coulombscattering wave function, for which the exact solution in parabolic coordinates is written in termsof the Kummer function. A power series expansion in terms of the Sommerfeld parameter isconsidered, and analytic closed form expressions up to order 2 were given for the terms, which



are functions only of the energy and the parabolic coordinate. Further studies of the hypergeo-metric functions n here introduced are necessary and this is the subject of our current investi-gations.

Finally, an investigation of the derivatives of the generalized hypergeometric functions pFqwith respect to its parameters would also be of interest. In particular, the 2F1 case can be of highinterest because of its application to the study of many physical problems.

ACKNOWLEDGMENTS

One of the authors G. Gasaneo thanks the support by PICTR 03/0437 of the ANPCYT, PGI24/F038 of the UNS Argentina, and PIP5595 of CONICET.

APPENDIX A: DEMONSTRATION OF FORMULA 28 FOR G2To get the explicit formula 28 for G2, we start with the differential equation 25 it satisfies:

z d2dz2 + b z ddz aG2 = 2G1, A1and the expression of G1 Eq. 17a:

G1 =z

b1

m1=0

m2=0

1m11m2am1a + 1m1+m2a + 1m12m1+m2b + 1m1+m2

zm1+m2

m1!m2!. A2

Using solution 13 of the inhomogeneous Kummer equation 12, we have

G2 =2

b1

m1=0

m2=0

1m11m2am1a + 1m1+m2a + 1m12m1+m2b + 1m1+m2

1m1!m2!

m1+m2+1+1a,b;z , A3

which in terms of hypergeometric functions 2F2 becomes

G2 =2

b1

m1=0

m2=0

1m11m2am1a + 1m1+m2a + 1m12m1+m2b + 1m1+m2

zm1+m2+2

m1!m2!

1

2 + m1 + m2b + 1 + m1 + m2 2F2 1,m1 + m2 + 1 + a + 1

m1 + m2 + 1 + 2,m1 + m2 + 1 + b + 1 ;z

A4

Using the two identities

1n +

=

1

n + 1n

,

m+n = m + mn,

and simplifying, we find

G2 =1

b1b + 11

m1=0

m2=0

1m11m2am1a + 1m1+m23m1+m2a + 1m1b + 2m1+m2

zm1+m2+2

m1!m2! 2F2 1,a + m1 + m2 + 2

m1 + m2 + 3,b + m1 + m2 + 2 ;z . A5

Now, replacing the series expansion index m3 for the 2F2 function leads to a triple series:



G2 =1

b1b + 11

m1=0

m2=0

m3=0

1m11m2am1a + 1m1+m23m1+m2a + 1m1b + 2m1+m2

1m3a + m1 + m2 + 2m3

m1 + m2 + 3m3b + m1 + m2 + 2m3

zm1+m2+m3+2

m1!m2!m3!, A6

which, after some algebraic manipulations, finally simplifies into

G2 =z2

b1b + 11

m1=0

m2=0

m3=0

1m11m21m33m1+m2+m3b + 2m1+m2+m3

am1a + 1m1+m2a + 2m1+m2+m3

a + 1m1a + 2m1+m2

zm1+m2+m3

m1!m2!m3!. A7

Similar calculations can be performed to obtain Gn for n2.

APPENDIX B: REDUCTION FORMULAS FOR n

In this appendix, we provide the reduction formulas for the hypergeometric functions ndefined by Eq. 34, in the case where b1 is set to zero. Starting from the n=1 case, we see that inthe double series expansion 19, only the m1=0 term survives in the summation, so that

1a1,a20,b2c1d1,d2

;x1,x2 = 2F2a2,b2d1,d2 ;x2 . B1Similarly from Eq. 33 for the n=2 case and from Eq. 34 for the general n1 case, one easilyfinds

2a1,a2,a30,b2,b3c1,c2d1,d2

;x1,x2,x3 =1a2,a3b2,b3c2d1,d2 ;x2,x3 , B2na1,a2, . . . ,an+10,b2, . . . ,bn+1

c1, . . . ,cnd1,d2 ;x1,x2, . . . ,xn+1

=n1a2, . . . ,an+1b2, . . . ,bn+1c2, . . . ,cnd1,d2

;x2, . . . ,xn+1 . B31 A. Erdelyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Trascendental Functions McGraw-Hill, New York,1953, Vols. IIII.

2 L. J. Slater, Confluent Hypergeometric Functions Cambridge University Press, London, 1960.3 H. Buchholz, The Confluent Hypergeometric Function Springer-Verlag, Berlin, 1969.4 L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory Pergamon, Oxford, 1965.5 C. H. Greene, A. R. P. Rau, and U. Fano, Phys. Rev. A 26, 2441 1982.6 A. Dzieciol, S. Yngve, and P. O. Frman, J. Math. Phys. 40, 6145 1999.7 C. R. Garibotti, G. Gasaneo, and F. D. Colavecchia, Phys. Rev. A 62, 022710 2000.8 L. U. Ancarani and M. C. Chidichimo, J. Phys. B 37, 4339 2004.9 H. van Haeringen, Charged Particle Interactions Coulomb, Leyden, 1985.

10 Y. E. Kim and A. L. Zubarev, Phys. Rev. A 56, 521 1997.11 R. Szmytkoswki, J. Phys. A 31, 4963 1998.12 G. Gasaneo and F. D. Colavecchia, J. Phys. A 36, 8443 2003.13 B. H. Bransden and C. J. Joachain, Physics of Atoms and Molecules, 2nd ed. Prentice Hall, Englewood Cliffs, NJ, 2003,

Chap. 13.14 M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions Dover, New York, 1972.15 J. Abad and J. Sesma, Comput. Phys. Commun. 156, 13 2003.16 A. W. Babister, Transcendental Functions Satisfying Nonhomogeneous Linear Differential Equations Macmillan, New

York, 1967.17 H. M. Srivastava and H. L. Manocha, A Treatise on Generating Functions Ellis Horwood, Chichester, 1978.



18 http://functions.wolfram.com/07.27.03.0119.0119 P. Appell and J. Kamp de Feriet, Funtions Hypergomtriques et Hypershriques; Polynomes dHermie Gauthier-

Villars, Paris, 1926.20 I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products Academic, New York, 1994.21 http://functions.wolfram.com/07.25.03.0076.01



derivadas de cualquier orden 1f1

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