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 an+1/an as n goes to infinity is a value \(0 \le L < 1\). The value of \(L\) is indicated in each entry. It is calculated automatically when the precompiled functions are used in the summation.
This series is the kernel of the Conway-Maxwell-Poisson distribution, which generalizes the Poisson and Geometric distributions. Its form is
$$L = 0, log(L) = -\infty,$$
for \(\lambda > 0\) and \(\nu > 0\).
When \(\nu = 1\), this series reduces to the Poisson distribution kernel and the sum (in the log scale) is known to be \(\lambda\). When \(\nu = 0\) and \(0 < \lambda < 1\), the series reduces to the Geometric distribution kernel with parameter \(1 - \lambda\). The series is known to sum to \(1\). Finally, as \(\nu\) goes to \(\infty\) the distribution approaches a Bernoulli distribution with parameter \(\lambda / (1 - \lambda)\).
Another known result is when \(\nu = 2\), in which case the sum is the modified Bessel function of the first kind of order 0 evaluated at \(2\sqrt{\lambda}\).
String to access the precompiled function: "COMP".
parameter vector: c(lambda, nu).
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
GP8hIamkYKW8vjk522fKO6UJeQQQJDLi$$L = 0, log(L) = -\infty,$$
for \(\lambda > 0\) and \(\phi > 0\).
When \(\phi = 1\), 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)
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
eIRg2OKn53IgRwNmD9X5651peTXEIwvW-1$$L = 0, log(L) = -\infty,$$
for \(x > 0\) and \(\alpha\) any real value.
The modified Bessel function of the second kind can be obtained with
where \(I\) represents the modified function of the first kind and \(K\)
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)
This is the same function as the one above, except that parameter \(x\) 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)
infiniteSum(), finiteSum() and
infiniteSum_batches()