aa r X i v : . [ m a t h - ph ] A ug Methods for Summing General Kapteyn Series
R. C. Tautz , I. Lerche , and D. Dominici E-mail: [email protected] , [email protected] , [email protected] Zentrum f¨ur Astronomie und Astrophysik, Technische Universit¨at Berlin,Hardenbergstraße 36, D-10623 Berlin, Germany Institut f¨ur Geowissenschaften, Naturwissenschaftliche Fakult¨at III,Martin-Luther-Universit¨at Halle, D-06099 Halle, Germany Department of Mathematics, State University of New York at New Paltz,1 Hawk Dr., New Paltz, NY 12561-2443, USA
Abstract.
The general features and characteristics of Kapteyn series, whichare a special type of series involving Bessel function, are investigated. For manyapplications to physics, astrophysics, and mathematics, it is crucial to have closed-form expressions in order to determine their functional structure and parametricbehavior. Closed-form expressions of Kapteyn series have mostly been limitedto special cases, even though there are often similarities in the approaches usedto reduce the series to analytically tractable forms. The goal of this paper is toreview the previous work in the area and to show that Kapteyn series can beexpressed as trigonometric or gamma function series, which can be evaluated inclosed form for specific parameters. Two examples with a similar structure aregiven, showing the complexity of Kapteyn series.
1. Introduction
Kapteyn series of the first and second kind have arisen in a large variety of physicsproblems since their discovery by Kapteyn (1893). Series of the first kind are of theform K ≡ K ( { a n } , α, β, c, b ) = ∞ X n = −∞ a n J αn + β ( cn + b ) (1)with α, β, c , and b fixed but possibly complex, and { a n } is a sequence of complexcoefficients. Series of the second kind are of the form K ≡ K ( { a n } , α, β, γ, ǫ, c, b, f, g )= ∞ X n = −∞ a n J αn + β ( cn + b ) J γn + ǫ ( f n + g ) (2)with α, β, γ, ǫ, c, b, f , and g fixed but possibly complex and { a n } is a sequence ofcomplex coefficients. Over the years, the subject of summing Kapteyn series hasattracted both physics (see below) and mathematics researchers (e. g., Erdelyi, 1981;Watson, 1966).One of the more important aspects of such Kapteyn series is to provide closedform expressions for the series in particular problems of physical interest. Thisaspect is often crucial, as one attempts to determine the functional structure and umming General Kapteyn Series N equallycharged particles uniformly spaced on a rotating ring, Budden (1924) used Kapteynseries to show how the far-field radiation distribution varies as N → ∞ (see alsoLerche and Tautz, 2008). Earlier, Schott (1912) had discussed radiation from a singlerelativistic particle moving in a circle. Both radiation problems involved Kapteynseries of the second kind with α = γ = 1; β = ǫ = 0; c = f and b = g = 0 but withdifferent coefficients a n for the two problems.To name but a few illustrations from a broad range of physical applications (seeTautz and Lerche, 2009, for a review), the large variety of physics problems involvingvarious Kapteyn series includes: (i) Kepler’s problem (Dattoli et al., 1998) (ii) pulsars(Harrison and Tademaru, 1975; Lerche and Tautz, 2007); (iii) side-band spectra oftera-hertz electromagnetic waves (Citrin, 1999; Lerche et al., 2009); (iv) high-intensityCompton scattering (Harvey et al., 2009; Lerche and Tautz, 2010); (v) queueingtheory (Dominici, 2007 b ); and (vi) cosmic ray transport theories (Tautz and Dominici,2010; Tautz and Lerche, 2010).In Dominici (2007 a ), Kapteyn series of the form K ( z, t ) = ∞ X n =1 t n J n ( nz ) (3)were studied and a series representation was derived in powers of z . Furthermore, theradius of convergence was analyzed. Also, general Kapteyn series of the first kind wereconsidered. In Dominici (2010), an asymptotic approximation in terms of a Kapteynseries was obtained for the zeros of the Hermite polynomials.As far as can be determined, physical applications of Kapteyn series to date seemto involve only structural behaviors of the form X a n J n + ν ( cn ) (4)for Kapteyn series of the first kind and X a n J n + β ( cn ) J − ( n + ǫ ) ( cn ) (5)for Kapteyn series of the second kind. What would be of value is to determinebroad ranges of the parameters and coefficients a n so that rather general closed-formrepresentations of the Kapteyn series K and K are available. Such knowledge wouldthen obviate having to evaluate each application of Kapteyn series de novo .Efforts in this general direction have been provided by Nielsen (1901, 1904) whosummed particular Kapteyn series of the second type. Curiously, in respect of Nielsen’swork, Watson (1966) remarks “series of the type X β n J ν + n (cid:20)(cid:18) µ + ν n (cid:19) z (cid:21) J µ + n (cid:20)(cid:18) µ + ν n (cid:19) z (cid:21) have been studied in some detail by Nielsen (1901). But the only series of this typewhich have, as yet, proved to be of practical importance, are some special series with µ = ν , and with simple coefficients.” However, Watson also goes on to say that“Schott (1912) has shown that” ∞ X n =1 J n ( nz ) = 12 h(cid:0) − z (cid:1) − / − i ;but direct inspection of equation (31) from Nielsen (1901) shows that the formulaascribed to Schott was already available. Indeed many further direct summations of umming General Kapteyn Series ∞ X n =1 J n [(2 n + 1) x ] J ′ n [(2 n + 1) x ] = 12 x h(cid:0) − x (cid:1) − / − i . (6)The difference in philosophy between Nielsen and Schott is that Nielsen treatedthe summation as a pure mathematics’ problem requiring summation, while Schottworked out the physics problem of synchrotron radiation involving the series. Thus,physics applications of the Kapteyn series arose a decade (or so) after the series wasoriginally summed in closed form.It would seem that to prejudge the ability to provide summations of as manyas possible Kapteyn series of the second kind as not of practical use is basically notappropriate, for applications often follow much later than the basic mathematicalresults.For these reasons, it seems relevant to consider de novo the series K and toattempt to determine procedures for evaluating such series for as broad a range ofparameters and coefficients as possible. In Secs. 2 and 3, general methods as well astwo specific examples for the summation of Kapteyn series of the second kind willbe presented, respectively. Sec. 4 provides a short summary and a discussion of theresults.
2. Methods for Summing K Series
In this section, different approaches are investigated that have proven useful forsumming various Kapteyn series of the second kind.
One of the main techniques for summing Kapteyn series of the second kind is (Erdelyi,1981; Gradshteyn and Ryzhik, 2000) J µ ( z ) J ν ( z ) = 2 π Z π/ d θ J µ + ν (2 z cos θ ) cos( µ − ν ) θ, (7)which is valid for µ, ν any integer values and is otherwise valid for ℜ ( µ + ν ) > − J µ and J ν have the same argument.Thus, when considering K ≡ ∞ X n = −∞ a n J αn + β ( cn + b ) J γn + ǫ ( f n + g ) (8)one restricts the evaluation using equation (7) to c = f and b = g . Then, for ℜ ( µ + ν ) > − ℜ [( α + γ ) n + β + ǫ ] > − n , with a n = 0.If a n = 0 for some integer n , then one requires ℜ ( α + γ ) = 0 and ℜ ( β + ǫ ) > − K = ∞ X n = −∞ a n J αn + β ( cn + b ) J − α R n +i γ I n + ǫ ( cn + b ) , (9)where γ = γ R + i γ I with γ R = − α R and α = α R + i α I , and ℜ ( β + ǫ ) > − umming General Kapteyn Series µ = − ν in equation (7), one has for the Kapteyn series ofthe second kind, K ( { a n } , α, β, − α, − β, c, b, c, b ), i. e., K = ∞ X n = −∞ a n J αn + β ( cn + b ) J − ( αn + β ) ( cn + b ) (10 a )= 2 π Z π/ ∞ X n = −∞ a n J [2 cos θ ( cn + b )] cos[2 θ ( αn + β )] . (10 b )Using the fact that J ( z ) = 1 π Z π d ψ cos( z sin ψ ) (11)one gets K = 2 π Z π/ d θ Z π d ψ ∞ X n = −∞ a n cos[2 θ ( αn + β )] cos[2 cos θ sin ψ ( cn + b )] (12 a )= 1 π Z π/ d θ Z π d ψ ∞ X n = −∞ a n × X j = ± j { cos[2 θ ( αn + β ) − j cos θ sin ψ ( cn + b )] } . (12 b )To the extent that one can sum expressions such as S = ∞ X n = −∞ a n cos( nA + B ) (13)in closed form, K can be, at worst, reduced to a double integral and, at best, canbe evaluated in closed form. The reduction depends precisely on the functional formschosen for a n and on the convergence of the terms in the series S (normally, but notnecessarily, by taking A to be real).For example, if a n = ( n + p ) − where p is not an integer, then one can write S = − pS ′ cos B + ∂S ′ ∂A sin B, (14)where S ′ = ∞ X n = −∞ cos nAn − p . (15)Because S ′ can be given in closed form, it is possible to sum a variety of Kapteynseries of the second kind with this procedure.Thus, for sideband spectra in the tera-hertz regime one can show (Lerche et al.,2009) that ∞ X ν =1 J ν + n ( aν ) J ν − n ( aν ) ν − b = ( − n − π b csc( πb ) J n + b ( ab ) J n − b ( ab ) (16)for n integer and n >
1, with 0 < a < < b < ∞ X ν =0 ( − n − ν (cid:0) ν + (cid:1) − b J n + ν +1 (cid:0) a ( ν + ) (cid:1) J n − ν (cid:0) a ( ν + ) (cid:1) = ( − n π b sec( πb ) J n + + b ( ab ) J n + − b ( ab ) (17) umming General Kapteyn Series n integer and n >
1, with 0 < a < < b < .Similarly, for high-intensity Compton scattering, series arise of the form ∞ X n =1 n p ( a + n ) q J n ( na ) (18)with p and q positive integers, which can be reduced to analytic closed form apart froma single elliptic integral that has to be added to the rest of the closed form expressions(Lerche and Tautz, 2010).Besides such evaluations where µ = − ν , the more general case with ℜ ( µ + ν ) > − µ = αn + β (19 a ) ν = − α R n + i γ I n + ǫ, (19 b )one obtains µ + ν = i( γ I + α I ) n + β + ǫ (20 a ) µ − ν = 2 α R n + i( α I − γ I ) n + β − ǫ. (20 b )Thus, one can write K = 2 π Z π/ d θ ∞ X n = −∞ a n J in ( γ I + α I )+ β + ǫ (cid:0) θ ( cn + b ) (cid:1) × cos (cid:2) θ (cid:0) α R n + i( α I − γ I ) n + β − ǫ (cid:1)(cid:3) (21)with ℜ ( β + ǫ ) > −
1. The cosine factors in equation (21) converge if and only if α I = γ I ,so that K = 2 π Z π/ d θ ∞ X n = −∞ a n J nα I + β + ǫ (cid:0) θ ( cn + b ) (cid:1) cos [ θ (2 α R n + β − ǫ ] . (22)Therefore, J ν ( x ) = 2 π Z x d t sin (cid:16) x cosh t − πν (cid:17) cosh νt, (23)for x ∈ R , so that c and b are real. But if ν contains an imaginary part proportional to n , as in equation (22), then the series in equation (22) diverges exponentially unless a n converges fast enough (e. g., a n proportional to exp[ − bn ]). Thus, one requires α I ≡ K = (cid:18) π (cid:19) Z ∞ d t Z π/ d θ ∞ X n = −∞ a n cos[ ν (2 α R n + β − ǫ )] × cosh (cid:2) ( β + ǫ ) t (cid:3) sin h θ ( cn + b ) cosh t − π β + ǫ ) i . (24)Again, one sees that for choices of a n such that the series in equation (24) can besummed in closed form then K is reduced, at worst, to a double integral and, at best,can be evaluated explicitly.All of these procedures for summing the general second-order Kapteyn seriesrepresented by K are dependent on the integral representation from equation (7) for J ν ( z ) J µ ( z ), valid for ℜ ( µ + ν ) > − µ, ν are not integer and otherwise generallyvalid.But just because there are values of ℜ ( µ + ν ) −
1, for which equation (7) cannotbe used, does not mean that other Kapteyn series of the K form cannot be summed.One needs other procedures to effect the summations when ℜ ( µ + ν ) − umming General Kapteyn Series - - Ν Figure 1.
The Kapteyn series from equation (28) for varying real ν ∈ [ − , x = 1 / The use of series representations to sum K types of Kapteyn series was already knownto Nielsen (1901) and later the same procedure was given by Watson (1966). FollowingNielsen (1901), the sense of the argument is as follows: One considers first integralsof the form I ν ( a ) = Z π/ d x cos ν − x cos ax (25)and, for ℜ ( ν ) >
0, expresses the result as I ν ( a ) = π cos aπ/ ν ν Γ( ν + 1)Γ (cid:0) ( ν + 1 + a ) / (cid:1) Γ (cid:0) ( ν + 1 − a ) / (cid:1) . (26)Then one considers equation (7) with µ = n − a and ν = n + a . According to Nielsen,one then uses the series expansion definition of the Bessel function under the integralsign as J α ( z ) = (cid:16) z (cid:17) α ∞ X k =0 ( − k k k ! z k Γ( α + k + 1) (27)and so one integrates equation (7) term by term with µ and ν as defined above byusing equation (27). Effectively, one trades a sum over Bessel functions of the Kapteynkind for a power series. The resulting power series can often, but not universally, beeither summed in closed form or can be evaluated for specific parameter values. Inthis way, Nielsen argued thatsin νπνπ + 2 ∞ X n =1 J n + ν (2 nx ) J n − ν (2 nx ) = 1 √ π ∞ X n =0 n ! Γ( n + )(2 x ) n Γ( n + 1 + ν )Γ( n + 1 − ν ) , (28)and2 ∞ X n =0 J n + ν [(2 n + 1) x ] J n +1 − ν [(2 n + 1) x ] = 1 √ π ∞ X n =0 n ! Γ( n + )(2 x ) n +1 Γ( n + 1 + ν )Γ( n + 2 − ν ) , (29) umming General Kapteyn Series - - Ν Figure 2.
The Kapteyn series from equation (29) for varying real ν ∈ [ − , x = 1 / which are illustrated in figures 1 and 2, respectively, for varying ν and for x = 1 / ν ∈ C and forcomplex x ∈ C with | x | < /
2. However, for ℑ ( ν ) >
1, both series attain extremelylarge values, depending on ℜ ( ν ) and x .Nielsen then commented that for ν = 0 one has the particular cases1 + 2 ∞ X n =1 J n (2 nx ) = (cid:0) − x (cid:1) − / (30)and 2 ∞ X n =0 J n [(2 n + 1) x ] J n +1 [(2 n + 1) x ] = 12 x h(cid:0) − x (cid:1) − / − i . (31)Note that the results shown here correct two misprints in Nielsen’s results [hisequations (30a) and (31)], which are: (i) Nielsen wrote J n − ν [(2 n + 1) x ] on the left-hand side of equation (29) instead of J n +1 − ν [(2 n + 1) x ] and (ii) Nielsen included theterm n = 0 when summing the left-hand side of equation (30).In fact, however, a more general representation is possible if, instead of usingNielsen’s series expansion of the Bessel function under the integral sign of equation (7),one were to write J n (2 nx cos θ ) = 1 π Z π d ψ cos 2 nψ cos(2 nx cos ψ ) (32) umming General Kapteyn Series ∞ X n =1 α n cos 2 nψ cos(2 nx cos ψ )for particular choices of α n . Nielsen’s choice of α n = 1 is just one example where thesummation can be achieved.Perhaps of more general interest is to ask how reducible equation (2) is forarbitrary parameter values. Then, using equation (27) for each of the Bessel functionsin K one has K = ∞ X n = −∞ a n ∞ X k =0 ∞ X r =0 (cid:18) cn + b (cid:19) αn + β ( − k ( cn + b ) k k k ! Γ( αn + β + k + 1) × (cid:18) f n + g )2 (cid:19) γn + ǫ ( − r ( f n + g ) r r r ! Γ( γn + ǫ + r + 1) . (33)Unless f n + g = Λ( cn + b ), where Λ is a constant, it is difficult to make further headwaywith equation (33). But when such is the case then one has K = ∞ X n = −∞ B n ∞ X k =0 ∞ X r =0 (cid:18) cn + b (cid:19) ( α + γ ) n + ǫ + β × ( − k ( − r k r k ! r ! ( cn + b ) k + r ) Λ r Γ( αn + β + k + 1)Γ( γn + ǫ + r + 1) , (34)where B n = a n Λ γn + ǫ . Now, setting k + r = m , one gets K = ∞ X n = −∞ B n (cid:18) cn + b (cid:19) ( α + γ ) n + ǫ + β ∞ X m =0 m X k =0 (cid:18) cn + b (cid:19) m × ( − m Λ m Λ − k k !( m − k )! Γ( αn + β + k + 1)Γ( γn + ǫ + 1 + m − k ) . (35)The representation of K in closed form (or at worst as an integral) then rests on theextent to which one can sum the various component sums occurring in equation (35).One can write K = ∞ X n = −∞ B n (cid:18) cn + b (cid:19) ( γ + α ) n + ǫ + β Q n , (36)where Q n = m X k =0 ∞ X m =0 (cid:18) cn + b (cid:19) m × ( − m Λ m Λ − k k !( m − k )! Γ( αn + β + k + 1)Γ( γn + ǫ + 1 + m − k ) . (37)Note that the replacement of r by ( m − k ) causes Q n to be a single power series in( cn + b ) / Q n = ∞ X m =0 (cid:18) cn + b (cid:19) m ( − m Λ m R m (38) umming General Kapteyn Series R m = ∞ X k =0 Λ k k !( m − k )! (cid:2) Γ( αn + β + k + 1) Γ( γn + ǫ + ( m − k ) + 1) (cid:3) − . (39)The basic question is: under what conditions is R m expressible in closed form? If itis, then one can then determine the conditions under which Q n is expressible in closedform and so arrange values of B n so that K is in closed form. It would seem thatonly for particular values of the parameters it is possible to effect closed-form results,as those for instance given by Nielsen [his equation (8)] (cid:16) x (cid:17) µ + ν = ( µ + ν ) Γ(1 + µ )Γ(1 + ν ) ∞ X n =0 (cid:18) µ + ν + n − n (cid:19) ( µ + ν + 2 n ) − ( µ + ν +1) × J µ + n [( µ + ν + 2 n ) x ] J ν + n [( µ + ν + 2 n ) x ] (40)
3. Examples for K series There are special cases of Kapteyn series of the second kind, which are often neededand which rely on other methods than those described above. One example is aKapteyn series, which consists of J n ( nz ) for z ∈ C with | z | <
1, combined with apower of n in the form K ≡ K (cid:0) n q , , , , , z, , z, (cid:1) = ∞ X n =1 n q J n ( nz ) . (41)Two different distinctions can be made: (i) q <
0; (ii) q >
0; each for (a) integer q ∈ N ; (b) arbitrary q ∈ R . q < p = − q so that K = ∞ X n =1 J n ( nz ) n p . (42)The general procedure is the following: use Bessel’s equation J n ( z ) , to show that (cf.Watson, 1966, equation 17.33), for consecutive indices p , the following two Kapteynseries of the first kind are related through the equation (cid:18) z dd z (cid:19) ∞ X n =1 J n (2 nz ) n p = 4 (cid:0) − z (cid:1) ∞ X n =1 J n (2 nz ) n p − . (43)The second initial condition, i. e., the sum for p = 1 (cf. Watson, 1966, Sec. 17.23), ∞ X n =1 J n (2 nz ) n = z , (44)together with Meissel’s (1892) investigation suggests one should write theKapteyn series of the first kind as a polynomial in z k (see Watson, 1966,Sec. 17.23). By evaluating the recurrence relation, equation (43), it has been shown(Tautz and Dominici, 2010) that the Kapteyn series of the first kind can be expressedas ∞ X n =1 J n (2 nz ) n p = p X k =1 z k k X j =1 ( − j + k j k − p ) ( k − j )!( k + j )! . (45) umming General Kapteyn Series p Ξ p Ξ p Ξ p Ξ Figure 3.
The functions ξ k ( p ) for varying p with k ∈ { , , , } as given throughequation (47) and, in explicit form, equation (49). The vertical axis should merelyillustrate the zeros of the functions ξ k ( p ), thus explaining why, for p integer, a finite power series is obtained in equation (47). To obtain the corresponding Kapteyn series of the second kind, equation (41),one employs equation (7) for µ = ν = n . Then one evaluates equation (45) with theargument 2 nz cos θ and integrates over θ , noting that (cf. Gradshteyn and Ryzhik,2000, Sec. 3.621)2 π Z π/ d θ cos n θ = (2 n − n )!! = Γ (cid:0) n + (cid:1) n ! √ π , n ∈ N , (46)where ( · )!! is the double factorial and where Γ( · ) denotes the Gamma function. Hence,the result is K ( z ) = Θ X k =1 z k Γ (cid:0) k + (cid:1) k ! √ π k − X j =0 ( − j ( k − j ) k − p ) j !(2 k − j )! , (47)with Θ = (cid:26) p , p ∈ N ∞ , p ∈ R \ N (48)and is valid for arbitrary p ∈ R < p , a sum with p ( p + 1) / p , an infinite power series occurs. But evenin that case, it is advantageous to exchange one infinite series (the Kapteyn series) byanother infinite series (a power series), because the convergence behavior of a powerseries is better understood, thus allowing for a more reliable estimate of the numberof terms needed to obtain a desired accuracy.The reason for the distinction between a finite/infinite power series is that, whenexpanding the coefficients of the powers z k , one finds non-algebraic functions ξ k ( p )of the form ξ ( p ) = −
116 + 4 − (1+ p ) umming General Kapteyn Series ξ ( p ) = 1768 (cid:0) − − p + 3 − p (cid:1) ξ ( p ) = 118 432 (cid:0) − − p + 7 · − p − − p (cid:1) ξ ( p ) = 12 949 120 (cid:0) − − p − · − p + 5 − p + 9 − p (cid:1) , (49)each of which has zeros at the first integers, i. e., ξ k = 0 for p = 1 , . . . , k −
1. Thefunctions ξ k ( p ) are illustrated in figure 3 for varying p and for k ∈ { , , , } . q > q > K ( z, q ) ≡ K (cid:0) n q , , , , , z, , z, (cid:1) = ∞ X n =1 n q J n (2 nz ) , (50)were investigated and it was found that K ( z, q ) = ∞ X n =1 " n !) n X k =0 ( − k (cid:18) nk (cid:19) ( n − k ) n + q ) z n , (51)for q > | z | < / . The result in equation (51) allows one to compute K ( z, q )numerically for any q > , ; however, sometimes it is difficult to use it to get closed-formexpressions. Here, a different approach will be introduced using differential operators.From Watson (1966, equation 5.4 (4)), it is known that the function y n ( t ) = J n (e t ) satisfies the differential equationd y n d t + 4 (cid:0) e t − n (cid:1) d y n d t + 4e t y n = 0 . (52)By changing variables to 2 nz = e t in equation (52), one obtains (cid:20) z d d z + 3 z d d z + (cid:0) n z − n + 1 (cid:1) dd z + 16 n z (cid:21) J n (2 nz ) = 0 , (53)or (cid:18) z d d z + 3 z d d z + dd z (cid:19) J n (2 nz ) = (cid:20) (cid:0) − z (cid:1) dd z − (cid:21) n J n (2 nz ) . (54)Introducing the function g q ( z, n ) = n q J n (2 nz ) , (55)one has (cid:18) z d d z + 3 z d d z + dd z (cid:19) g q = (cid:20) (cid:0) − z (cid:1) dd z − (cid:21) g q +1 , (56)for all n ∈ N ⋆ , i. e., all positive integers.Thus, it follows that the function K ( z, q ) satisfies (cid:18) z d d z + 3 z d d z + dd z (cid:19) K ( z, q ) = (cid:20) (cid:0) − z (cid:1) dd z − (cid:21) K ( z, q + 1) , (57)while equation (2) gives for the initial condition K ( z,
0) = −
12 + 12 √ − z . (58) umming General Kapteyn Series K H z,q L Figure 4.
The first functions K ( z, q ) for varying z ∈ [0 , /
2] with q ∈ { , , , } as given through equation (62) and, in explicit form, equation (61). The order q varies from right to left, i. e., the solid line shows K (1 , z ) while the dotted lineshows K (4 , z ). Since J n (0) = 0 for all n = 1 , , . . . , one has K (0 , q ) = 0. Solving equation (57) for K ( z, q + 1) , one obtains K ( z, q + 1) = 14 √ − z z Z d w √ − w (cid:18) w d d w + 3 w d d w + dd w (cid:19) K ( w, q ) , (59)where one must be careful that | z | < / K in the form of a the recurrence relation, which reads K ( z, q + 1) = " z − z ) d d z + z (cid:0) − z (cid:1) − z ) dd z + 2 z (cid:0) z (cid:1) (1 − z ) K ( z, q ) − √ − z z Z d w w (cid:0) w + 4 w (cid:1) (1 − w ) / K ( w, q ) . (60)Using equation (60) and (58), it is straightforward (albeit tedious) to compute theexplicit expressions for the first orders of the series K , yielding the relations K ( z,
1) = z (cid:0) z (cid:1) (1 − z ) / K ( z,
2) = z (cid:0) z + 118 z + 27 z (cid:1) (1 − z ) / K ( z,
3) = z (cid:0) z + 5036 z + 23 630 z + 22 910 z + 2250 z (cid:1) (1 − z ) / K ( z,
4) = z (cid:0)
385 875 z + 7 119 756 z + 15 359 862 z + 8 635 578 z + 1 515 705 z + 80130 z + 973 z + 1 (cid:1) (cid:0) − z (cid:1) − / , (61) umming General Kapteyn Series K ( z, q ) = z P q (cid:0) z (cid:1) (1 − z ) q +1 / , | z | < , (62)where P q ( z ) is a polynomial of degree 2 q −
1. The structure of the polynomials P q ( z )is quite complicated and will be analyzed in a forthcoming paper.
4. Discussion and Conclusion
In this article, the general features and characteristics of Bessel function series wereinvestigated. Special emphasis was focused on Kapteyn series, which appear inmany applications of theoretical physics and mathematics, such as radiation andoptimization problems. In their original form, with the index of summation appearingin both the index and the argument of the Bessel function(s) involved, the convergenceof such series is, in general, unclear. Therefore, it is appropriate and necessary toundertake every effort of rewriting such sums in terms of, at worst, infinite power seriesor (double) integrals, the convergence of which can be estimated more reliably. Moreimportantly, in many cases it has proven possible to find closed analytical expressionsfor Kapteyn series of both the first and the second kind. This is indispensable for caseswhere Kapteyn series constitute only part of large mathematical expressions that haveto be dealt with numerically.However, the quest for closed-form expressions of Kapteyn series has mostly beenlimited to special cases that have arisen in specific problems such as those listed inSec. 1. Often one notes that a similar procedure proves useful for different forms ofKapteyn series, even in such cases where the summation coefficients are considerablydiverse. But no general answer has been found to date to the following problem: Forwhich parameter regimes of the coefficients do the Kapteyn series have a closed-formexpression?It was the aim of the present article to shed some light on that question.Starting from the most general form of Kapteyn series of the second kind, i. e.,involving the product of two Bessel functions, the Kapteyn series was decomposedinto series over trigonometric functions or, more generally, algebraic expressionsinvolving gamma functions. The ability to sum such series depends on the precisechoice of the parameters. However, it has been shown that the likelihood for anyanalytical tractability is increased if both Bessel functions are of equal order, i. e.,( αn + β ) = ( γn + ǫ ) in equation (2).Two specific examples with applications, for instance, in cosmic ray diffusiontheories (Tautz and Dominici, 2010; Shalchi and Schlickeiser, 2004; Tautz et al., 2006)were illustrated, where the summation coefficients are simply powers of the summationindex. It has been shown that, depending very sensitively on the parameters chosenfor the summation coefficients, power series with a finite/infinite number of terms areobtained, where the relative magnitude of each term can now easily be estimated.Future work should, presumably, concentrate on the application of one or moreof the above methods (or indeed combinations of the methods) to determine to themaximum extent possible the broadest range of conditions for summability of thegeneral Kapteyn series. Of course, new methods for attempting such summations arewelcome and it would be highly interesting to see any such ideas that would add tothe capability to effect Kapteyn series summations. Another interesting question is to umming General Kapteyn Series Acknowledgments
The authors are thankful to the digitalization services of several online archives,without which the access to the work of mathematicians around the beginning of the20 th Century would be considerably more difficult. The work of D. Dominici waspartially supported by a Humboldt Research Fellowship for Experienced Researchersfrom the Alexander von Humboldt Foundation.
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