expxisumj

Expxisumj is a theoretical construct used in mathematics and related disciplines to denote the exponential of the sum of a sequence indexed by j. It serves as a compact notation for combining accumulation of values with an exponential transformation.

Definition and notation

For a finite sequence j1, j2, ..., jk, expxisumj is defined as exp(∑_{i=1}^k ji). If the sequence is infinite but the series converges, expxisumj is defined as exp(∑_{i=1}^∞ ji). The concept emphasizes the order-aggregation of contributions before applying the exponential function.

Variants and interpretation

The idea can be applied in contexts where the sum represents log-weights, log-probabilities, or cumulants. If ji = log(ai) for i = 1,...,k, then expxisumj = ∏_{i=1}^k ai. In probabilistic and combinatorial settings, expxisumj often appears in the study of partition functions and generating functions.

Properties

Expxisumj inherits standard exponential properties: exp(a+b) = exp(a) exp(b) and exp(ca) = (exp(c))^a when c is a scalar. The operation is monotone with respect to the underlying sum, and its magnitude grows with the total accumulated input.

Applications

Expxisumj is used as a compact formalism in analyses of partition functions, statistical mechanics, and generating functions. It helps simplify expressions that would otherwise involve repeated exponentials after summation.

Computation and stability

In numerical work, computing exp(∑ ji) directly can lead to overflow or underflow. The log-sum-exp technique is often employed to improve stability, evaluating the sum in log-space before exponentiation.