Let be even with and nonzero. Then
In the following two identities, valid for , the sums are taken over all nonnegative integers with .
In the next identity, valid for , the sum is taken over all positive integers with .
For (24.14.11) and (24.14.12), see Al-Salam and Carlitz (1959). These identities can be regarded as higher-order recurrences. Let denote a Hankel (or persymmetric) determinant, that is, an determinant with element in row and column for . Then
See also Sachse (1882).