The Fourier-Legendre Series of Bessel Functions of the First Kind

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The Fourier-Legendre Series of Bessel Functions of the First Kind

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A peer-reviewed article of this preprint also exists.

Jack C. Straton^*

This version is not peer-reviewed

Submitted:

25 December 2023

Posted:

26 December 2023

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Abstract

The Bessel function of the first kind J_N(kx) is expanded in a Fourier-Legendre series, as is the modified Bessel functions of the first kind I_N(kx) . Known polynomial approximations for the range −3≤x≤3 (having five-digit accuracy) are compared with those arising from this Legendre series and substitute versions are given with twice the number of terms and range (with five-digit accuracy or having fifteen-digit accuracy over −3≤x≤3). It is shown that infinite series of like-powered contributors (involving _2F_3 hypergeometric functions) extracted from the Fourier-Legendre series may be summed, having values that are inverse powers of the first five primes 1/(2^i 3^j 5^k 7^l 11^mm) for powers of order x24.

Keywords:

Subject: Computer Science and Mathematics - Applied Mathematics

1. Introduction

The Strong Field Approximation (SFA) [1,2,3,4,5,6], unlike perturbation expansions that will not converge if an applied laser field is large, is an analytical approximation that is nonperturbative. Keating [7] applied it to the production of the positive antihydrogen ion and found that the reduction of the transition amplitudes to analytic form required the expansion of the resultant Bessel functions

J_{N} (k x)

in a series of spherical harmonics. We have been unable to discover such a Laplace series [8] in the literature, nor does there seem to be an expansion of the Bessel function in a series of Legendre polynomials, to which the Laplace series reduces in the common case where the function is independent of the azimuthal angle. We recreate herein Keating’s derivation, stripped of the specialized SFA terminology, and find that we can recast his series as a ₂F₃ generalized hypergeometric function. We also extend the method to the modified Bessel function of the first kind

I_{N} (k x)

Since a Fourier-Legendre series is easily converted to a series in powers, we do so and compare the results to the known polynomial approximation[9] for the range

- 3 \leq x \leq 3

(having five-digit accuracy). We easily generate a substitute version with twice the number of terms that gives twice the range (with five-digit accuracy or having fifteen-digit accuracy over

- 3 \leq x \leq 3

). Using 48-digit accuracy, we show that like-powered contributors extracted from the Fourier-Legendre series may be summed, having values that are inverse powers of the first five primes

1 / (2^{i} 3^{j} 5^{k} 7^{l} 11^{m})

for powers of order

x^{24}

2. The Fourier-Legendre series of a Bessel function of the first kind

We begin with the assumption that the series

J_{N} (k x) = \sum_{L = 0}^{\infty} a_{L N} P_{L} (x)

(1)

converges uniformly[10], where the coefficients are given by the orthogonality of the Legendre polynomials,

a_{L N} = \frac{2 L + 1}{2} \int_{- 1}^{1} J_{N} (k x) P_{L} (x) d x

(2)

Following Keating’s lead, we use Heine’s integral representation of the Bessel function [11] for integer indices

J_{N} (k x) = \frac{i^{- N}}{π} \int_{0}^{π} e^{i k x cos θ} cos (N θ) d θ,

(3)

so that

a_{L N} = \frac{2 L + 1}{2} \int_{- 1}^{1} [\frac{i^{- N}}{π} \int_{0}^{π} e^{i k x cos θ} cos (N θ) d θ] P_{L} (x) d x .

(4)

By switching the order of integration,

a_{L N} = \frac{2 L + 1}{2} \frac{i^{- N}}{π} \int_{0}^{π} [\int_{- 1}^{1} e^{i k x cos θ} P_{L} (x) d x] cos (N θ) d θ,

(5)

we can use [12,13]

e^{i k x cos θ} = \sum_{l^{'}} (2 l^{'} + 1) i^{l^{'}} j_{l^{'}} (k cos θ) P_{l^{'}} (x) .

(6)

Then

\begin{matrix} a_{L N} & = & \frac{2 L + 1}{2} \frac{i^{- N}}{π} \int_{0}^{π} [\int_{- 1}^{1} (\sum_{l^{'} = 0}^{\infty} (2 l^{'} + 1) i^{l^{'}} j_{l^{'}} (k cos θ) P_{l^{'}} (x)) P_{L} (x) d x] cos (N θ) d θ \\ = & \frac{2 L + 1}{2} \frac{i^{- N}}{π} \int_{0}^{π} [(\sum_{l^{'} = 0}^{\infty} (2 l^{'} + 1) i^{l^{'}} j_{l^{'}} (k cos θ) \frac{2}{2 l^{'} + 1} δ_{l^{'} L})] cos (N θ) d θ \\ = & (2 L + 1) \frac{i^{L - N}}{π} \int_{0}^{π} j_{L} (k cos θ) cos (N θ) d θ . \end{matrix}

(7)

Using the series expansion [14]

j_{l} (x) = \frac{1}{2} \sqrt{π} {(\frac{x}{2})}^{l} \sum_{M = 0}^{\infty} \frac{{(- \frac{1}{4})}^{M} x^{2 M}}{M! Γ (l + M + \frac{3}{2})}

(8)

this becomes

\begin{matrix} a_{L N} & = & \frac{(2 L + 1)}{\sqrt{π}} i^{L - NN} 2^{- L - 1} (\sum_{M = 0}^{\infty} \frac{{(- \frac{1}{4})}^{M} k^{L + 2 M}}{M! Γ (L + M + \frac{3}{2})}) \int_{0}^{π} {cos}^{L + 2 M} (θ) cos (N θ) d θ . \end{matrix}

(9)

Gröbner and Hofreiter[15] extended an integral over the interval

[0, \frac{π}{2}]

that has three branches, to the interval

[0, π]

with a prefactor

(1 + {(- 1)}^{m + n})

that renders the central one of the three possibilities nonzero only for even values for

m + n

\int_{0}^{π} c o s^{m} θ c o s (n θ) d θ = (1 + {(- 1)}^{m + n}) \frac{π}{2^{m + 1}} (\binom{m}{\frac{m - n}{2}}) [m \geq n > - 1, m - n = 2 K] .

(10)

The other two branches, being for odd

m + n

[0, \frac{π}{2}]

, are zero on

[0, π]

when this prefactor is included. (Numerical integration also confirms that the contributions from the

[\frac{π}{2}, π]

interval cancels the contributions from the

[0, \frac{π}{2}]

on these branches.) The fifth edition of Gradshteyn and Ryzhik[16] (in which

m ⟷ n

) nevertheless included all three branches and this prefactor on the interval

[0, π]

. By their seventh edition, Gradshteyn and Ryzhik removed this integral entirely despite the correctness of the central branch on the interval

[0, π]

. Neither source noted the lower limit on m that we found:

[m \geq n > - 1]

The final form for the coefficient set of the Fourier-Legendre series for the Bessel function

J_{N} (k x)

is then

\begin{matrix} a_{L N} & = & \sqrt{π} (2 L + 1) 2^{- L - 1} i^{L - N} \sum_{M = 0}^{\infty} \frac{({(- \frac{1}{4})}^{M} k^{L + 2 M})}{2^{L + 2 M + 1} (M! Γ (L + M + \frac{3}{2}))} (1 + {(- 1)}^{L + 2 M + N}) (\binom{L + 2 M}{\frac{1}{2} (L + 2 M - N)}) \\ = & \frac{\sqrt{π} 2^{- 2 L - 2} (2 L + 1) k^{L} i^{L - N}}{Γ (\frac{1}{2} (2 L + 3))} (1 + {(- 1)}^{L + N}) (\binom{L}{\frac{L - N}{2}}) \\ \times & _{2} F_{3} (\frac{L}{2} + \frac{1}{2}, \frac{L}{2} + 1; L + \frac{3}{2}, \frac{L}{2} - \frac{N}{2} + 1, \frac{L}{2} + \frac{N}{2} + 1; - \frac{k^{2}}{4}) \end{matrix},

(11)

where the final step is new with the present work.

The first 13 terms in the sum (1) are then, to 48-digit accuracy with

k = 1

(using

E - 7

as the programming shorthand for

\times 10^{- 7})

\begin{matrix} J_{0} (x) & ≅ & 0.919730410089760239314421194080619970661964806513 P_{0} (x) \\ - & 0.157942058625851887573713967144363701344798627015 P_{2} (x) \\ + & 0.00343840094460110923299688787207291548407319575920 P_{4} (x) \\ - & 0.0000291972184882872969366059098612566327416694039900 P_{6} (x) \\ + & 1.317356952447780977655616563143279878442358000327244788969743 E - 7 P_{8} (x) \\ - & 3.6845008442082030271737710960588661990942840123477263963 E - 10 P_{10} (x) \\ + & 7.0118300329938459282088033282114472785242458071919272574718378083 E - 13 P_{12} (x) \\ - & 9.6659643698589122636719953727533464513227149569317835997999035 E - 16 P_{14} (x) \\ + & 1.00963627682454644652534217092493625236608417531220118935645 E - 18 P_{16} (x) \\ - & 8.2666569559276378589919725841741165080506716379747942367136935206 E - 22 P_{18} (x) \\ + & 5.4482448677627587258900828378394303583525504857936407629268043 E - 25 P_{20} (x) \\ - & 2.9525271821373547516757746066633996448458337416324847736021 E - 28 P_{22} (x) \\ + & 1.3388561588585344690808986700962001092415532661176110513810834869 E - 31 P_{24} (x) . \end{matrix}

(12)

When compared to the polynomial approximation given in Abramowitz and Stegun,[9]

\begin{matrix} J_{0} (x) ≅ 1 - 2.25 {(\frac{x}{3})}^{2} + 1.26562 {(\frac{x}{3})}^{4} - 0.316387 {(\frac{x}{3})}^{6} + 0.0444479 {(\frac{x}{3})}^{8} - 0.0039444 {(\frac{x}{3})}^{10} \\ + 0.0002100 {(\frac{x}{3})}^{12}, \end{matrix}

(13)

at the latter’s limiting value of

x = 3

for five-digit accuracy, one finds 15-digit accuracy in the result,

- 0.260051954901933

, much better than one would expect for doubling the number of powers in the approximation. In addition, the range of applicability of the new Legendre approximation truncated at

x^{24}

, if one is satisfied with five-digit accuracy, roughly doubles to

J_{0} (6.5) ≅ 0.260095

3. Summing a set of infinite series

If the original polynomial approximation [9] has its powers of three folded into the coefficients,

\begin{matrix} J_{0} (x) ≅ 1 - 0.25 x^{2} + 0.0156249 x^{4} - 0.000434001 x^{6} + 6.77456 E - 6 x^{8} - 6.6799 E - 8 x^{10} \\ + 3.952 E - 10 x^{12} . \end{matrix}

(14)

and notes that the third term is

1 / 64.0004

, one is led to wonder if each of these coefficients is made up of inverse powers of primes if one were to rederive a similar approximation using a process with greater accuracy.

We can easily expand the Legendre polynomials into their constituents terms and gather like powers in eq. (12) to give an updated polynomial approximation,

\begin{matrix} J_{0} (x) & ≅ & 1.00000000000000000000000000000000000000000000000 x^{0} \\ - & 0.250000000000000000000000000000000000000000000000 x^{2} \\ + & 0.0156250000000000000000000000000000000000000000000 x^{4} \\ - & 0.000434027777777777777777777777777777777777777777778 x^{6} \\ + & 6.7816840277777777777777777777777777777777777777777709035743914 E - 6 x^{8} \\ - & 6.78168402777777777777777777777777777777777777776418460183472 E - 8 x^{10} \\ + & 4.709502797067901234567901234567901234567901217102330430492 E - 10 x^{12} \\ - & 2.40280754952443940539178634416729654824891336675938319116842797541 E - 12 x^{14} \\ + & 9.3859669903298414273116654069035021414920770461703578216582006 E - 15 x^{16} \\ - & 2.896903392077111551639402903365278385973981447743521706878477 E - 17 x^{18} \\ + & 7.242258480192778879098507258412992706519050135221465476326 E - 20 x^{20} \\ - & 1.4963343967340452229542370368012108220961398389215474513676266228 E - 22 x^{22} \\ + & 2.5978027721077174009622169341723075518748354228202179137433269 E - 25 x^{24} . \\ = & 1 x^{0} - \frac{x^{2}}{2^{2}} + \frac{x^{4}}{2^{6}} - \frac{x^{6}}{2^{8} 3^{2}} + \frac{x^{8}}{2^{14} 3^{2}} - \frac{x^{10}}{2^{16} 3^{2} 5^{2}} + \frac{x^{12}}{2^{20} 3^{4} 5^{2}} - \frac{x^{14}}{2^{22} 3^{4} 5^{2} 7^{2}} + \frac{x^{16}}{2^{30} 3^{4} 5^{2} 7^{2}} \\ - & \frac{x^{18}}{2^{32} 3^{8} 5^{2} 7^{2}} + \frac{x^{20}}{2^{36} 3^{8} 5^{4} 7^{2}} - \frac{x^{22}}{2^{38} 3^{8} 5^{4} 7^{2} 11^{2}} + \frac{x^{24}}{2^{44} 3^{10} 5^{4} 7^{2} 11^{2}} . \end{matrix}

(15)

We see that we have managed to rederive the first thirteen terms of the well-known series representation[17]

J_{ν} (x) = \sum_{k = 0}^{\infty} \frac{{(- 1)}^{k} {(\frac{x}{2})}^{2 k + ν}}{k! Γ (k + ν + 1)}

(16)

But the outcome worth the trouble of this investigation is that this process yields a set of infinite sums whose values are inverse powers of primes. Looking back at the coefficients in eq. (12) when multiplied by the constant terms in the Legendre polynomials that multiply them, whose first few are

\begin{matrix} \{P_{0} (x) = 1, P_{2} (x) = \frac{1}{2} (3 x^{2} - 1), P_{4} (x) = \frac{1}{8} (35 x^{4} - 30 x^{2} + 3), \\ P_{6} (x) = \frac{1}{16} (231 x^{6} - 315 x^{4} + 105 x^{2} - 5)\}, \end{matrix}

(17)

there is no reason to suspect that

\begin{matrix} 0.919730410089760239314421194080619970661964806513 \\ - \frac{1}{2} (- 0.157942058625851887573713967144363701344798627015) \\ + \frac{3}{8} 0.00343840094460110923299688787207291548407319575920 \\ - \frac{5}{16} (- 0.0000291972184882872969366059098612566327416694039900) \\ + \dots & = & 1 \end{matrix}

(18)

but one sees uniform convergence up through to accuracy of the calculation as one adds additional terms, as seen in Table 1.

One may more formally concluded that

\sum_{L = 0}^{\infty}^{(2)} \frac{\sqrt{π} i^{L} (2 L + 1) (\binom{L}{\frac{L}{2}}) (\binom{2 L}{L}) {(\frac{1}{2} - \frac{L}{2})}_{\frac{L}{2}} {(- \frac{L}{2})}_{\frac{L}{2}} 2^{- 3 L - 2}_{1} F_{2} (\frac{L}{2} + \frac{1}{2}; \frac{L}{2} + 1, L + \frac{3}{2}; - \frac{1}{4})}{\frac{L}{2}! {(\frac{1}{2} - L)}_{\frac{L}{2}} Γ (\frac{1}{2} (2 L + 3))} = 1

(19)

where the superscript “(2)” on the sum indicates one is summing even values only (or one my retain the factor

(1 + {(- 1)}^{L})

in the sum), which is a result we have not seen in the literature. The Pochhammer symbols

{(\frac{1}{2} - \frac{L}{2})}_{\frac{L}{2}}

and so on derive from a shift to[18]

P_{n} (x) = 2^{- n} (\binom{2 n}{n}) x^{n}_{2} F_{1} (\frac{1}{2} - \frac{n}{2}, - \frac{n}{2}; \frac{1}{2} - n; \frac{1}{x^{2}})

(20)

in the explicit sum. [19] One can see from the repeating digits that the third through sixth lines in eq. (15) have inverses that are powers of primes, and one can even see that the fifth term is

2^{2} 5^{2}

times the sixth.

Subsequent terms are not at all obviously inverse powers of primes. Indeed, that eventuality requires a sufficient number of terms even at our 48-digit accuracy. It turns out that the inverse of the coefficient of the

x^{16}

term when including the 13 terms in the sum (15), that were sufficient for checking convergence of coefficients of the smallest powers, is

106542032486495.616348409991752462411671619456197,

whose integer part is in bold face. This is not a product of low-level primes. Adding one more term to (15) is sufficient to bring it to

106542032486400.113376300998684305400416345779209

, whose integer part is

2^{30} 3^{4} 5^{2} 7^{2}

. Adding another term gives

106542032486400.000104784167278249059923631013442

and with every additional term added, the integer part remains the same while the fractional part diminishes by several decimal places. The coefficient of the

x^{24}

term required 40 terms in the sum (15) to establish convergence, and all of the coefficients in (15) include contributions from all 40 terms to establish a consistent floating-point set as well as the inverse prime version. One could likely use these as seed values for a 13-term optimization attempt that would be slightly more accurate than the present truncation approach, but simply adding more terms to a truncated series is much easier given the analytic form at our disposal in eq. (11).

The other 12 verified summed series given by the present approach (with

1 \leq h \leq 12

) are

\begin{matrix} \sum_{L = 0}^{\infty}^{(2)} & \frac{\sqrt{π} i^{L} 2^{- 3 L - 2} ({(- 1)}^{L} + 1) (2 L + 1) (\binom{L}{\frac{L}{2}}) (\binom{2 L}{L})_{1} F_{2} (\frac{L}{2} + \frac{1}{2}; \frac{L}{2} + 1, L + \frac{3}{2}; - \frac{1}{4})}{Γ (\frac{1}{2} (2 L + 3)) (\frac{L}{2} - h)!} \\ \times & [\frac{{(\frac{1}{2} - \frac{L}{2})}_{\frac{L}{2} - h} {(- \frac{L}{2})}_{\frac{L}{2} - h}}{{(\frac{1}{2} - L)}_{\frac{L}{2} - h}}] = \frac{{(- 1)}^{h} 2^{- 2 h}}{h! Γ (h + 1)}, \end{matrix}

(21)

from which one may verify the primes in the last two lines of eq. (15).

One may also use the relation[20]

{(- a)}_{n} = {(- 1)}^{n} {(a - n + 1)}_{n}

(22)

and the primary definition of the Pochhammer symbol[21]

{(a)}_{n} = \frac{Γ (a + n)}{Γ (a)} /; \neg (- a \in Z \land - a \geq 0 \land n \in Z \land n \leq - a)

(23)

to rewrite the term in square brackets in infinite sum (21) to give

\begin{matrix} \sum_{L = 0}^{\infty}^{(2)} & \frac{\sqrt{π} i^{L} 2^{- 3 L - 2} ({(- 1)}^{L} + 1) (2 L + 1) (\binom{L}{\frac{L}{2}}) (\binom{2 L}{L})_{1} F_{2} (\frac{L}{2} + \frac{1}{2}; \frac{L}{2} + 1, L + \frac{3}{2}; - \frac{1}{4})}{Γ (\frac{1}{2} (2 L + 3)) (\frac{L}{2} - h)!} \\ \times & [\frac{{(- 1)}^{\frac{L}{2} - h} {(h + \frac{1}{2})}_{\frac{L}{2} - h} {(h + 1)}_{\frac{L}{2} - h}}{{(h + \frac{L}{2} + \frac{1}{2})}_{\frac{L}{2} - h}}] \\ = & \sum_{L = 0}^{\infty}^{(2)} \frac{\sqrt{π} i^{L} 2^{- 3 L - 2} ({(- 1)}^{L} + 1) (2 L + 1) (\binom{L}{\frac{L}{2}}) (\binom{2 L}{L})_{1} F_{2} (\frac{L}{2} + \frac{1}{2}; \frac{L}{2} + 1, L + \frac{3}{2}; - \frac{1}{4})}{Γ (\frac{1}{2} (2 L + 3)) (\frac{L}{2} - h)!} \\ \times & [\frac{{(- 1)}^{\frac{L}{2} - h} 2^{2 h - L} Γ (L + 1) Γ (h + \frac{L}{2} + \frac{1}{2})}{Γ (2 h + 1) Γ (L + \frac{1}{2})}] = \frac{{(- 1)}^{h} 2^{- 2 h}}{h! Γ (h + 1)} . \end{matrix}

(24)

4. Series arising from the $J_{1} (x)$ Fourier-Legendre series

The first 13 terms in the

J_{1} (x)

Fourier-Legendre series (1) are

\begin{matrix} J_{1} (x) & = & 0.463598170595381063594111003933870249258071550272 P_{1} (x) \\ - & 0.0238653456584073979630720941648486611447567718049 P_{3} (x) \\ + & 0.000319724355972004763852475762325602811615940515016 P_{5} (x) \\ - & 1.9705191806665942502580629293911116487585393730843186079480334 E - 6 P_{7} (x) \\ + & 6.9872474730978072187917594101570139463303280482950147787763 E - 9 P_{9} (x) \\ - & 1.610500056046875027807002442953327025382453835843708565744143862464 E - 11 P_{11} (x) \\ + & 2.607086592441628842939248193619909317796212820945088296037258834 E - 14 P_{13} (x) \\ - & 3.127311482540796882144713619567442440315805193381169051898182 E - 17 P_{15} (x) \\ + & 2.89142408178705073982738259661606356990165332364232995931 E - 20 P_{17} (x) \\ - & 2.123664534779369199214414455720317392205378180144888812801656722 E - 23 P_{19} (x) \\ + & 1.269011201758673511714553707528185762589997449465032454145276 E - 26 P_{21} (x) \\ - & 6.290201939135925763576871358738600140562742682788804579147609808134 E - 30 P_{23} (x) \\ + & 2.628135796989325452573870774267213117030660982177348165961027847 E - 33 P_{25} (x) . \end{matrix}

(25)

When compared to the polynomial approximation given in Abramowitz and Stegun [22],

\begin{matrix} J_{1} (x) & ≅ & \frac{x}{2} - 0.062499983 x^{3} + 0.0026041448 x^{5} - 0.00005424265 x^{7} + 6.7568816 E - 7 x^{9} \\ - & 5.3788 E - 9 x^{11} + 2.087 E - 11 x^{13}, \end{matrix}

(26)

at the latter’s limiting value of

x = 3

for five-digit accuracy, one finds 16-digit accuracy in the result,

0.3390589585259365

, again much better than one would expect for doubling the number of powers in the approximation. In addition, the range of applicability of the new Legendre approximation, if one is satisfied with five-digit accuracy, roughly doubles to

J_{1} (6.5) ≅ - 0.153841

We can again expand the Legendre polynomials into their constituents terms and gather like powers in eq. (12) to give an updated polynomial approximation,

\begin{matrix} J_{1} (x) & ≅ & 0.499999999999999999999999999999999996073036732708 x \\ - & 0.0624999999999999999999999999999995064877689984265 x^{3} \\ + & 0.00260416666666666666666666666664830697662637912184 x^{5} \\ - & 0.0000542534722222222222222222219048407193034541315431 x^{7} \\ + & 6.781684027777777777777746918775410200512596641822542207329666 E - 7 x^{9} \\ - & 5.6514033564814814814627959321470018420114848588327454860019 E - 9 x^{11} \\ + & 3.363930569334215160073367965443021892055161549117566544419386429142 E - 11 x^{13} \\ - & 1.5017547184527725848458902100520196712921972004803304061942047588 E - 13 x^{15} \\ + & 5.2144261057349232956805346216422729234612607233549650639373514 E - 16 x^{17} \\ - & 1.44845169552820912740631826455853618243090549911868532723491 E - 18 x^{19} \\ + & 3.29193521573211266128200143559435222243437141235203698749095423421 E - 21 x^{21} \\ - & 6.23446088053356596553816078985229136419498303017048139336940432 E - 24 x^{23} \\ + & 9.90105390245274625404949366616978333091783081722258589870395 E - 27 x^{25} \\ = & \frac{x}{2} - \frac{x^{3}}{2^{4}} + \frac{x^{5}}{2^{7} 3} - \frac{x^{7}}{2^{11} 3^{2}} + \frac{x^{9}}{2^{15} 3^{2} 5} - \frac{x^{11}}{2^{18} 3^{3} 5^{2}} + \frac{x^{13}}{2^{21} 3^{4} 5^{2} 7} + \frac{x^{15}}{2^{26} 3^{4} 5^{2} 7^{2}} + \frac{x^{17}}{2^{31} 3^{6} 5^{2} 7^{2}} \\ - & \frac{x^{19}}{2^{34} 3^{8} 5^{3} 7^{2}} + \frac{x^{21}}{2^{37} 3^{8} 5^{4} 7^{2} 11} - \frac{x^{23}}{2^{41} 3^{9} 5^{4} 7^{2} 11^{2}} + \frac{x^{24}}{2^{45} 3^{10} 5^{4} 7^{2} 11^{2} 13}, \end{matrix}

(27)

again the first thirteen terms of the well-known series representation[17]. The bridge from the Legendre series to the above gives us another set of infinite series (of which we explicate

0 \leq h \leq 12

, above),

\begin{matrix} \sum_{L = 1}^{\infty}^{(2)} & \frac{\sqrt{π} i^{L - 1} ({(- 1)}^{L + 1} + 1) (2 L + 1) 2^{- 3 L - 2} (\binom{L}{\frac{L - 1}{2}}) (\binom{2 L}{L}) {(- 1)}^{- h + \frac{L}{2} - \frac{1}{2}}_{1} F_{2} (\frac{L}{2} + 1; \frac{L}{2} + \frac{3}{2}, L + \frac{3}{2}; - \frac{1}{4})}{Γ (\frac{1}{2} (2 L + 3)) (- h + \frac{L}{2} - \frac{1}{2})!} \\ \times & [\frac{{(\frac{1}{2} - \frac{L}{2})}_{\frac{L - 1}{2} - h} {(- \frac{L}{2})}_{\frac{L - 1}{2} - h}}{{(\frac{1}{2} - L)}_{\frac{L - 1}{2} - h}}] \\ = & \sum_{L = 1}^{\infty}^{(2)} \frac{\sqrt{π} i^{L - 1} ({(- 1)}^{L + 1} + 1) (2 L + 1) 2^{- 3 L - 2} (\binom{L}{\frac{L - 1}{2}}) (\binom{2 L}{L})_{1} F_{2} (\frac{L}{2} + 1; \frac{L}{2} + \frac{3}{2}, L + \frac{3}{2}; - \frac{1}{4})}{Γ (\frac{1}{2} (2 L + 3)) (- h + \frac{L}{2} - \frac{1}{2})!} \\ \times & [{(- 1)}^{- h + \frac{L}{2} - \frac{1}{2}} \frac{{(h + 1)}_{- h + \frac{L}{2} - \frac{1}{2}} {(h + \frac{3}{2})}_{- h + \frac{L}{2} - \frac{1}{2}}}{{(h + \frac{L}{2} + 1)}_{- h + \frac{L}{2} - \frac{1}{2}}}] \\ = & \sum_{L = 1}^{\infty}^{(2)} \frac{\sqrt{π} i^{L - 1} ({(- 1)}^{L + 1} + 1) (2 L + 1) 2^{- 3 L - 2} (\binom{L}{\frac{L - 1}{2}}) (\binom{2 L}{L})_{1} F_{2} (\frac{L}{2} + 1; \frac{L}{2} + \frac{3}{2}, L + \frac{3}{2}; - \frac{1}{4})}{Γ (\frac{1}{2} (2 L + 3)) (- h + \frac{L}{2} - \frac{1}{2})!} \\ \times & [{(- 1)}^{- h + \frac{L}{2} - \frac{1}{2}} \frac{2^{2 h - L + 1} Γ (L + 1) Γ (h + \frac{L}{2} + 1)}{Γ (2 h + 2) Γ (L + \frac{1}{2})}] = \frac{{(- 1)}^{h} 2^{- 2 h - 1}}{h! Γ (h + 2)} . \end{matrix}

(28)

5. Series arising from the $I_{n} (x)$ Fourier-Legendre series

Because the modified Bessel functions of the first kind

I_{N} (k x)

are related to the ordinary Bessel functions by the relation [23]

I_{n} (z) = i^{- n} J_{n} (i z),

(29)

we merely need to multiply by

i^{- n}

and set

k = i

in eq. (11) to obtain the

I_{0} (x)

Fourier-Legendre series, the first 13 terms of which are

\begin{matrix} I_{0} (x) & ≅ & 1.086521097023589815837941923492506345739351342949 P_{0} (x) \\ + & 0.175804681921524266260595135426125001605976541743 P_{2} (x) \\ + & 0.00370900924405288253392383816552703258834979223948 P_{4} (x) \\ + & 0.0000309510527099243219861374460877760164076890739973 (x) \\ + & 1.381259734719773538320052305224505914245982116085251360367698 E - 7 P_{8} (x) \\ + & 3.83431260108637300531778890612557340476082424326359129125 E - 10 P_{10} (x) \\ + & 7.2571724500962139367206676604119779116847154554922929445210425096 E - 13 P_{12} (x) \\ + & 9.9627469788360180201284331116359751150453844926209747761545488 E - 16 P_{14} (x) \\ + & 1.03725134611005263096370547704673552658307922276998581986119 E - 18 P_{16} (x) \\ + & 8.470496863240475339343499321604115510305543113181010728012323846 E - 22 P_{18} (x) \\ + & 5.5705413998588522192782604835236866603601422817930129556765446 E - 25 P_{20} (x) \\ + & 3.0133473832345288502246890418232012922500194443178483562736 E - 28 P_{22} (x) \\ + & 1.36433800535352727263847909317524941305304452973661274851014159257 E - 31 P_{24} (x) . . \end{matrix}

(30)

When compared to the polynomial approximation given in Abramowitz and Stegun,[24]

\begin{matrix} I_{0} (x) & ≅ & 1 + 3.5156229 {(\frac{x}{3.75})}^{2} + 3.0899424 {(\frac{x}{3.75})}^{4} + 1.2067492 {(\frac{x}{3.75})}^{6} \\ + & 0 . . 2659732 {(\frac{x}{3.75})}^{8} + 0.0360768 {(\frac{x}{3.75})}^{10} + 0.0045813 {(\frac{x}{3.75})}^{12} \\ = & 1 + 0.25 x^{2} + 0.0156252 x^{4} + 0.00043394 x^{6} + 6.801234 E - 6 x^{8} + 6.56017 E - 8 x^{10} \\ + & 5.9240 E - 10 x^{12} \end{matrix}

(31)

at the latter’s limiting value of

x = 3.75

for seven-digit accuracy, one finds 15-digit accuracy in the result,

9.118945860844

, about what one would expect for doubling the number of powers in the approximation. In addition, the range of applicability of the new Legendre approximation, if one is satisfied with six-digit accuracy, roughly doubles to

I_{0} (7.5) ≅ 268.1613

. It is only in the latter form of the polynomial approximation, in comparison with the

J_{0}

version (14) that one would suspect that the two are related by reversing all of the negative signs in the

J_{0}

version. That the correspondence is not exact for the higher-power terms likely is a result of the optimization scheme in the two cases having slightly different ranges of validity.

We may, however, use our truncation method to verify that the two series are identical apart from the sign reversals of the terms containing

i^{- 2 m}

multiplied by

x^{2 m}

for m odd. The full series [25] indeed fulfills this correspondence:

I_{ν} (x) = \sum_{k = 0}^{\infty} \frac{{(\frac{x}{2})}^{2 k + ν}}{k! Γ (k + ν + 1)} .

(32)

We thereby have another set of summed series given by our bridge from the

I_{0} (x)

Fourier-Legendre series for

h \geq 0

\begin{matrix} \sum_{L = 0}^{\infty}^{(2)} & \frac{\sqrt{π} i^{L} k^{L} 2^{- 3 L - 2} ({(- 1)}^{L} + 1) (2 L + 1) (\binom{L}{\frac{L}{2}}) (\binom{2 L}{L})_{1} F_{2} (\frac{L}{2} + \frac{1}{2}; \frac{L}{2} + 1, L + \frac{3}{2}; - \frac{k^{2}}{4})}{Γ (\frac{1}{2} (2 L + 3)) (\frac{L}{2} - h)!} \\ \times & {[\frac{{(\frac{1}{2} - \frac{L}{2})}_{\frac{L}{2} - h} {(- \frac{L}{2})}_{\frac{L}{2} - h}}{{(\frac{1}{2} - L)}_{\frac{L}{2} - h}}]}_{k \to i} \\ = & \sum_{L = 0}^{\infty}^{(2)} \frac{\sqrt{π} i^{2 L} 2^{- 3 L - 2} ({(- 1)}^{L} + 1) (2 L + 1) (\binom{L}{\frac{L}{2}}) (\binom{2 L}{L})_{1} F_{2} (\frac{L}{2} + \frac{1}{2}; \frac{L}{2} + 1, L + \frac{3}{2}; \frac{1}{4})}{Γ (\frac{1}{2} (2 L + 3)) (\frac{L}{2} - h)!} \\ \times & [\frac{{(\frac{1}{2} - \frac{L}{2})}_{\frac{L}{2} - h} {(- \frac{L}{2})}_{\frac{L}{2} - h}}{{(\frac{1}{2} - L)}_{\frac{L}{2} - h}}] \\ = & \sum_{L = 0}^{\infty}^{(2)} \frac{\sqrt{π} i^{2 L} 2^{- 3 L - 2} ({(- 1)}^{L} + 1) (2 L + 1) (\binom{L}{\frac{L}{2}}) (\binom{2 L}{L})_{1} F_{2} (\frac{L}{2} + \frac{1}{2}; \frac{L}{2} + 1, L + \frac{3}{2}; \frac{1}{4})}{Γ (\frac{1}{2} (2 L + 3)) (\frac{L}{2} - h)!} \\ \times & [\frac{2^{2 h - L} Γ (L + 1) Γ (h + \frac{L}{2} + \frac{1}{2})}{Γ (2 h + 1) Γ (L + \frac{1}{2})}] = \frac{2^{- 2 h}}{h! Γ (h + 1)} . \end{matrix}

(33)

Indeed, since

J_{υ} (z)

is well defined for complex z, we may develop a summed series using this method for any complex value of k. For instance, for

k \to \frac{1 + i}{\sqrt{2}}

one has

\begin{matrix} \sum_{L = 0}^{\infty}^{(2)} & \frac{\sqrt{π} i^{L} k^{L} 2^{- 3 L - 2} ({(- 1)}^{L} + 1) (2 L + 1) (\binom{L}{\frac{L}{2}}) (\binom{2 L}{L})_{1} F_{2} (\frac{L}{2} + \frac{1}{2}; \frac{L}{2} + 1, L + \frac{3}{2}; - \frac{k^{2}}{4})}{Γ (\frac{1}{2} (2 L + 3)) (\frac{L}{2} - h)!} \\ \times & {[\frac{{(\frac{1}{2} - \frac{L}{2})}_{\frac{L}{2} - h} {(- \frac{L}{2})}_{\frac{L}{2} - h}}{{(\frac{1}{2} - L)}_{\frac{L}{2} - h}}]}_{k \to \frac{1 + i}{\sqrt{2}}} = \frac{{(- 1)}^{h} {(1 + i)}^{2 h} 2^{- 3 h}}{h! Γ (h + 1)} \\ = & \{1, - \frac{i}{2^{2}}, - \frac{1}{2^{6}}, \frac{i}{2^{8} 3^{2}}, \frac{1}{2^{14} 3^{2}}, - \frac{i}{2^{16} 3^{2} 5^{2}}, - \frac{1}{2^{20} 3^{4} 5^{2}}, \dots\} for [h = 0, 1, 2, \dots] . \end{matrix}

(34)

Although eq. (1) allows one to easily compute the Fourier-Legendre series for any

J_{n} (x)

I_{n} (x)

, to enable readers to find these series for higher indices by recursion [26] we give the first 13 terms in the

I_{1} (x)

Fourier-Legendre series to complete the required set:

\begin{matrix} I_{1} (x) & = & 0.538634342185255559280908105166633575604956905216 P_{1} (x) \\ + & 0.0261806916482597744979529640726033318363969866342 P_{3} (x) \\ + & 0.000341985191255080623621009436150734413918978648008 P_{5} (x) \\ + & 2.0776519716996569638602670707248637741802282324973820998109474 E - 6 P_{7} (x) \\ + & 7.2990015186624314149055763249328768942564072680398941990154 E - 9 P_{9} (x) \\ + & 1.671443482954853739162527767203214630786040706834885109168930276284 E - 11 P_{11} (x) \\ + & 2.69274455145923523273493666645270433308765677977212169939443154 E - 14 P_{13} (x) \\ + & 3.218106754771162455853759838545281955092295174007919112773609 E - 17 P_{15} (x) \\ + & 2.966624646773403074824196435937541740846303862377661036696 E - 20 P_{17} (x) \\ + & 2.173686883720901031568436047655747923862489651656737239793224472 E - 23 P_{19} (x) \\ + & 1.296326265789554875546711294998344406345357835150626218271828 E - 26 P_{21} (x) \\ + & 6.41485510215141573359658829657883299044774009667838336427 E - 30 P_{23} (x) \\ + & 2.67638992514287578686307428538719583021215218577608245660976892 E - 33 P_{25} (x) . \end{matrix}

(35)

Because of the above-noted correspondence between the power-series versions of

J_{0} (x)

and

I_{0} (x)

(reversing all of the negative signs in the former to achieve the latter), there is no need to display the

I_{1} (x)

power-series version either since the same correspondence applies. We simply give the last set of infinite series,

\begin{matrix} \sum_{L = 1}^{\infty}^{(2)} & \frac{\sqrt{π} i^{L - 2} k^{L} 2^{- 3 L - 2} ({(- 1)}^{L + 1} + 1) (2 L + 1) (\binom{L}{\frac{L - 1}{2}}) (\binom{2 L}{L})_{1} F_{2} (\frac{L}{2} + 1; \frac{L}{2} + \frac{3}{2}, L + \frac{3}{2}; - \frac{k^{2}}{4})}{Γ (\frac{1}{2} (2 L + 3)) (\frac{L - 1}{2} - h)!} \\ \times & {[\frac{{(\frac{1}{2} - \frac{L}{2})}_{\frac{L - 1}{2} - h} {(- \frac{L}{2})}_{\frac{L - 1}{2} - h}}{{(\frac{1}{2} - L)}_{\frac{L - 1}{2} - h}}]}_{k \to i} \\ = & \sum_{L = 1}^{\infty}^{(2)} \frac{\sqrt{π} i^{2 L - 2} 2^{- 3 L - 2} ({(- 1)}^{L + 1} + 1) (2 L + 1) (\binom{L}{\frac{L - 1}{2}}) (\binom{2 L}{L})_{1} F_{2} (\frac{L}{2} + 1; \frac{L}{2} + \frac{3}{2}, L + \frac{3}{2}; \frac{1}{4})}{Γ (\frac{1}{2} (2 L + 3)) (\frac{L - 1}{2} - h)!} \\ \times & [\frac{{(- 1)}^{\frac{L - 1}{2} - h} {(h + 1)}_{\frac{L - 1}{2} - h} {(h + \frac{3}{2})}_{\frac{L - 1}{2} - h}}{{(h + \frac{L}{2} + 1)}_{\frac{L - 1}{2} - h}}] \\ = & \sum_{L = 1}^{\infty}^{(2)} \frac{\sqrt{π} {(- 1)}^{L - 1} 2^{- 3 L - 2} ({(- 1)}^{L + 1} + 1) (2 L + 1) (\binom{L}{\frac{L - 1}{2}}) (\binom{2 L}{L})_{1} F_{2} (\frac{L}{2} + 1; \frac{L}{2} + \frac{3}{2}, L + \frac{3}{2}; \frac{1}{4})}{Γ (\frac{1}{2} (2 L + 3)) (\frac{L - 1}{2} - h)!} \\ \times & [{(- 1)}^{\frac{L - 1}{2} - h} \frac{2^{2 h - L + 1} Γ (L + 1) Γ (h + \frac{L}{2} + 1)}{Γ (2 h + 2) Γ (L + \frac{1}{2})}] = \frac{2^{- 2 h - 1}}{h! Γ (h + 2)} . \end{matrix}

(36)

6. Conclusions

We have found the Fourier-Legendre series of modified Bessel functions of the first kind

I_{N} (k x)

based on that found by Keating [7] for the Bessel functions of the first kind

J_{N} (k x)

, and show that Keating’s coefficients, comprised of infinite-series, can be reduced to

_{2} F_{3}

functions. For

N = 0

and 1 we give numerical values for those coefficients up through

x^{24}

with 48-digit accuracy, and find that the resultant approximation gives 10 more decimal places of accuracy that the polynomial approximations given in Abramowitz and Stegun [9,22,24] or a roughly double their

- 3 \leq x \leq 3

range at their five-digit accuracy.

Each of these infinite Fourier-Legendre series may be decomposed into an infinite sum of infinite series, by gathering like powers from the Legendre polynomials in each of the terms in the Fourier-Legendre series. We show that each of these infinite sub-series converges to values that are inverse powers of the first five primes

1 / (2^{i} 3^{j} 5^{k} 7^{l} 11^{m})

for powers up to

x^{24}

. That these values recapitulate the coefficients of the known power series expansions of Bessel functions[17,25] is no surprise since we expect such an expansion to be unique. But given the relative paucity of infinite series whose values are known (e.g., two dozen pages in Gradshteyn and Ryzhik compared to their 900 pages of known integrals), having even one such to add to the total has the potential to be of use to future researchers. We add an infinite set of infinite series of

_{2} F_{3}

functions whose values are now known.

Funding

This research received no external funding.

Conflicts of Interest

The author declares no conflicts of interest.

References

“Effect of an intense electromagnetic filed on a weakly bound system,” H. R. Reiss. Phys. Rev. A 22, 1786 (1980). [CrossRef]
“Theoretical methods in quantum optics: S-matrix and Keldysh techniques for stong-field process,” H. R. Reiss. Progress in Quantum Electronics 16, 1 (1992).
“Relativistic strong-field photoionization,” H. R. Reiss. Journal of the Optical Society of America B 7, 4 p. 570 (1990). [CrossRef]
“Multiple absorbtion of laser phtons by atoms,” F. H. M. Faisal. J. Phys. B. vol. 6, 4 (1973). [CrossRef]
“Foundations of Strong-Field Physics,” Howard R. Reiss. Lecutres on Ultrafast intense Laser Scinece 1, 94 (2010) 41.
“Limits on Tunneling Theories of Strong-Field Ionization,” H. R. Reiss. Phys. Rev. Lett. 101, 043002 (2008). [CrossRef]
C. M. Keating, PhD Dissertation, “Using Strong Laser Fields to Produce Antihydrogen Ions,” appendix D,https://pdxscholar.library.pdx.edu/open_access_etds/4519/.
Howard, E. Haber, The Spherical Harmonics, http://scipp.ucsc.edu/~haber/ph116C/SphericalHarmonics_12.pdf.
Handbook of mathematical functions with formulas, graphs, and mathematical tables, ed M. Abramowitz and I. A. Stegun (National Bureau of Standards, Applied mathematics series, 55, Washington, 1970), p. 369 No. 9.4.1.
Oliver, D. Kellogg, Foundations of Potential Theory (Springer-Verlag, Berlin, 1929), p 132.
W. Magnus, F. W. Magnus, F. Oberhettinger, and R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics (Springer-Verlag, NY, 1966). p. 81; https://functions.wolfram.com/03.01.07.0005.01.
Gradshteyn, I.S.; Ryzhik, I.M. Table of Integrals, Series, and Products, 5th ed. (Academic: New York, NY, USA, 1994), p. 987 No. 8.511.4.
Joachain, C. J. Quantum Collision Theory; North-Holland, NY, 1983; p. 671, Eq. (B.46).
http://functions.wolfram.com/03.21.06.0019.01.
W. Gröbner und N. Hofreiter, Integraltafel Zweiter Teil Bestimmte Integrale (Springer-Verlag, Berlin, 1961), p. 110 No. 332.14a.
Supra note 12, p. 417 No. 3.631.17.
Supra note 12, p. 970 No. 8.440.
Prudnikov, A.P., Brychkov, Yu. A., and Marichev, O.I. Integrals and Series; (Gorden and Breach, New York, 1986) Vol. 3 p. 468 No. 7.3.1.206.
Ibid, p. 430 No. 7.2.1.1.
Relation 5.2.6 at https://dlmf.nist.gov/search/search?q=pochhammer&p=0&r=0.
http://functions.wolfram.com/06.10.02.0001.01.
Supra note 9, p. 370 No. 9.4.4.
Supra note 12, p. 961 No. 8.406.3.
Supra note 9, p. 378 No. 9.8.1.
Supra note 12, p. 971 No. 8.445.
Supra note 12, p. 979 No. 8.471.1; p. 981 No. 8.486.1.
Supra note 12, pp. 8-14, 42-44, 46-49, 54, 59-60.

Table 1. The constant term of the polynomial approximation as increasing numbers of terms are added from eq. (12), to 48 digit accuracy.

0.919730410089760239314421194080619970661964806513
0.998701439402686183101278177652801821334364120020
0.999990839756911599063652010604829164640891568430
0.999999963887689188843944699951660807338623340119
0.999999999909168357337955807722426205787682516277
0.999999999999841620300892054094280728854756176645
0.99999999999999979732605041136082528291421094660
0.999999999999999999801573588581204243621535893434
0.999999999999999999999846581786952424267000275450
0.999999999999999999999999903953851991585994488442
0.999999999999999999999999999950320420042897103088
0.999999999999999999999999999999978412290228685090
0.999999999999999999999999999999999992008312509362
0.999999999999999999999999999999999999997449396440
0.999999999999999999999999999999999999999999290955
0.999999999999999999999999999999999999999999999827
1.00000000000000000000000000000000000000000000000

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The Fourier-Legendre Series of Bessel Functions of the First Kind

Abstract

1. Introduction

2. The Fourier-Legendre series of a Bessel function of the first kind

3. Summing a set of infinite series

4. Series arising from the J 1 x Fourier-Legendre series

5. Series arising from the I n x Fourier-Legendre series

6. Conclusions

Funding

Conflicts of Interest

References

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4. Series arising from the $J_{1} (x)$ Fourier-Legendre series

5. Series arising from the $I_{n} (x)$ Fourier-Legendre series