Up to this point could be any strictly positive real number. From now on, however, we have to assume that is a strictly positive integer. In order to further simplify the expression obtained above, in general it will be necessary to write in terms of , which can be done using the recurrence formula of the Bessel functions [6], so long as is an integer. Let us examine a few of the initial cases. For we have simply
(36) |
so that the formula for the sum that corresponds to this case is
(37) |
were we used the properties of the gamma function, thus obtaining a polynomial on in the denominator. In this way we obtain the first of the known results, and this formula is therefore proven, being valid for any non-negative real value of . For we have
(38) |
In order to simplify this expression we write the recurrence formula as
(39) |
where we exchanged for . Applying this for and using once more the fact that , we get
(40) |
so that in this case we have
(41) |
from which it follows that the formula for the sum that corresponds to this case is
(42) |
We thus obtain the second known result, which is now proven, and which also has a polynomial on in the denominator. The proof of the first two formulas is therefore quite straightforward. In the case, however, something slightly different happens. In this case we have
(43) |
and hence we must write one more version of the recurrence formula. Exchanging for in the original formula, and applying at , we obtain
(44) |
Substituting in this the solution found for the previous case, which gives us in terms of , we get
(45) |
In this way we get in this case the result
(46) |
We see that in this case a linear combination of the sums of two different powers of the zeros appears. Using the properties of the gamma function and substituting the value obtained previously for , we get for
(47) |
The formula for the sum that corresponds to this case is therefore
(48) |
where we once more have a polynomial on in the denominator. We thus obtain the third of the known results, and the formula for is proven.
It is clear that we can proceed in this way indefinitely, thus obtaining the formulas for successive values of . In each case it is necessary to first use the recurrence formula in order to write in terms of . In general the result will be a linear combination of sums of several distinct inverse powers of . At this point the use of the general formula in Eq. (35) will produce an expression for the linear combination of the corresponding sums . Finally, it is necessary to solve the resulting expression for the sum with the largest value of so far, using for this end the results obtained previously for the other sums. In this way all the formulas for the sums can be derived successively by purely algebraic means, resulting every time in the ratio of two polynomials, with the one in denominator completely factored.