sumR: Approximate Summation of Series

#' List of precompiled functions in the sumR package
#'
#' More functions are periodically added to this list for convenience and speed.
#' These functions are all evaluated in the log scale and pass the ratio test,
#' that is, the limit of
#' \ifelse{html}{\out{a<sub>n+1</sub>/a<sub>n</sub>}}{\eqn{a_{n+1}/a_n}} as n
#' goes to infinity is a value \eqn{0 \le L < 1}. The value of \eqn{L} is
#' indicated in each entry. It is calculated automatically when the precompiled
#' functions are used in the summation.
#'
#' @note In some cases, the sum of the series is known in closed form for some
#' values of the parameters. The package function does not check for these cases
#' and just performs the approximation. If the exact value is desired by the
#' user when it is known, they must take responsibility for checking and
#' providing these values.
#'
#' Another important thing to note is that the precompiled functions perform all
#' calculations with twice the numerical precision than R. Therefore, in some
#' cases, there might be very small differences in the sum when comparing the
#' results of the function using the precompiled function and the same function
#' defined at the R level.
#'
#' @section Conway-Maxwell-Poisson normalizing constant:
#' This series is the kernel of the Conway-Maxwell-Poisson distribution, which
#' generalizes the Poisson and Geometric distributions. Its form is
#'
#' \ifelse{html}{\out{<div style='text-align: center;'>a<sub>n</sub> = λ<sup>n</sup> / (n!)<sup>ν</sup></div> }}{\deqn{a_n = \frac{\lambda^n}{(n!)^\nu},}}
#'
#' \deqn{L = 0, log(L) = -\infty,}
#'
#' for \eqn{\lambda > 0} and \eqn{\nu > 0}.
#'
#' When \eqn{\nu = 1}, this series reduces to the Poisson distribution kernel
#' and the sum (in the log scale) is known to be \eqn{\lambda}. When
#' \eqn{\nu = 0} and \eqn{0 < \lambda < 1}, the series reduces to the Geometric
#' distribution kernel with parameter \eqn{1 - \lambda}. The series is known to
#' sum to \eqn{1}. Finally, as \eqn{\nu} goes to \eqn{\infty} the distribution
#' approaches a Bernoulli distribution with parameter
#' \eqn{\lambda / (1 - \lambda)}.
#'
#' Another known result is when \eqn{\nu = 2}, in which case the sum is the
#' modified Bessel function of the first kind of order 0 evaluated at
#' \eqn{2\sqrt{\lambda}}.
#'
#' - String to access the precompiled function: `"COMP"`.
#'
#' - parameter vector: `c(lambda, nu)`.
#'
#' @section Double Poisson normalizing constant:
#' This series is the kernel of the double Poisson distribution, which is a
#' special case of the double exponential family, which extends it. Its form is
#'
#' \ifelse{html}{\out{<div style='text-align: center;'>a<sub>n</sub> = φ<sup>0.5</sup> e<sup>-φ µ</sup> (e<sup>-n</sup>n<sup>n</sup> / n!)(eµ / n)<sup>φ n</sup></div> }}{\deqn{a_n = \sqrt{\phi}e^{-\phi\mu}\left(\frac{e^{-n}n^n}{n!}\right)\left(\frac{e\mu}{n}\right)^{\phi n},}}
#'
#' \deqn{L = 0, log(L) = -\infty,}
#'
#' for \eqn{\lambda > 0} and \eqn{\phi > 0}.
#'
#' When \eqn{\phi = 1}, this series reduces to the Poisson distribution kernel
#' and the sum (in the log scale) is known to be 0.
#'
#' - String to acccess the precompiled function: `"double_poisson"`.
#'
#' - parameter vector: `c(mu, phi)`
#'
#' @section Modified Bessel function of the first kind:
#' This is the series form solution for the function. There are more time
#' efficient methods for its evaluation however they don't guarantee good
#' approximations with large parameters. Its form is
#'
#' \ifelse{html}{\out{<div style='text-align: center;'>a<sub>n</sub> = (x / 2)<sup>2k+α</sup> / (k!Γ(k + α + 1))</div> }}{\deqn{a_n = \left(\frac{x}{2}\right)^{2k+\alpha}\frac{1}{k!\Gamma(k+\alpha+1)},}}
#'
#' \deqn{L = 0, log(L) = -\infty,}
#'
#' for \eqn{x > 0} and \eqn{\alpha} any real value.
#'
#' The modified Bessel function of the second kind can be obtained with
#'
#' \ifelse{html}{\out{<div style='text-align: center;'> K<sub>α</sub>(x) = π / 2 (I<sub>-α</sub>(x) - I<sub>α</sub>(x)) / (sin α π)</div> }}{\deqn{K_\alpha(x) = \frac{\pi}{2}\frac{I_{-\alpha}(x)-I_\alpha(x)}{\sin\quad\alpha\pi},}}
#'
#' where \eqn{I} represents the modified function of the first kind and \eqn{K}
#' of the second kind. It is worth remembering the [infiniteSum()]
#' function returns the sum in the log scale, which must be adjusted for the
#' formula above.
#'
#' - String to access the precompiled function: `"bessel_I"`
#'
#' - parameter vector: `c(x, alpha)`
#' 
#' @section Modified Bessel function of the first kind with log argument:
#' This is the same function as the one above, except that parameter \eqn{x} is
#' given in the log scale. This is provided for numerical stability. For the
#' cases where `x` is not very large, sums using this function and the
#' above should return the same sum.
#' 
#' - String to access the precompiled function: `"bessel_I_logX"`
#'
#' - parameter vector: `c(logx, alpha)`
#'
#' @name precompiled
#' @seealso [infiniteSum()], [finiteSum()] and
#' [infiniteSum_batches()]
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