Let $(p_k)_{k=0, \dots, \infty}$ denote the probability masses of a Poisson distribution with parameter $\lambda$. I'm looking for the sum of their squares, $$\sum_{k=0}^\infty p_k^2,$$ as a function of $\lambda$. In other words I am interested in (the exponential of) the second-order Renyi entropy of a Poisson distribution.

Background:

• I'd like to use this to evaluate Brier and spherical scores.

• Czado et al. (2009) write that this expression can be evaluated analytically, but don't give further information, and I'm kind of stuck.

• No, this is not homework, although I imagine it could be ;-) Any hints or pointers to literature would be almost as much appreciated as a full solution.

• Here is the analogous question for the Negative Binomial distribution.

Let $(p_k)_{k=0, \dots, \infty}$ denote the probability masses of a Poisson distribution with parameter $\lambda$. I'm looking for the sum of their squares, $$\sum_{k=0}^\infty p_k^2,$$ as a function of $\lambda$. In other words I am interested in (the exponential of) the second-order Renyi entropy of a Poisson distribution.

Background:

• I'd like to use this to evaluate Brier and spherical scores.

• Czado et al. (2009) write that this expression can be evaluated analytically, but don't give further information, and I'm kind of stuck.

• No, this is not homework, although I imagine it could be ;-) Any hints or pointers to literature would be almost as much appreciated as a full solution.

• Here is the analogous question for the Negative Binomial distribution.

Let $(p_k)_{k=0, \dots, \infty}$ denote the probability masses of a Poisson distribution with parameter $\lambda$. I'm looking for the sum of their squares, $$\sum_{k=0}^\infty p_k^2,$$ as a function of $\lambda$.

Background:

• I'd like to use this to evaluate Brier and spherical scores.

• Czado et al. (2009) write that this expression can be evaluated analytically, but don't give further information, and I'm kind of stuck.

• No, this is not homework, although I imagine it could be ;-) Any hints or pointers to literature would be almost as much appreciated as a full solution.

• Here is the analogous question for the Negative Binomial distribution.

Let $(p_k)_{k=0, \dots, \infty}$ denote the probability masses of a Poisson distribution with parameter $\lambda$. I'm looking for the sum of their squares, $$\sum_{k=0}^\infty p_k^2,$$ as a function of $\lambda$. In other words I am interested in (the exponential of) the second-order Renyi entropy of a Poisson distribution.

Background:

• I'd like to use this to evaluate Brier and spherical scores.

• Czado et al. (2009) write that this expression can be evaluated analytically, but don't give further information, and I'm kind of stuck.

• No, this is not homework, although I imagine it could be ;-) Any hints or pointers to literature would be almost as much appreciated as a full solution.

• Here is the analogous question for the Negative Binomial distribution.