Applicability of standard forward column/row recurrence equations for ALFs

Fully normalized associated Legendre functions (fnALFs) are a set of orthogonal basis functions that are usually calculated by using the recurrence equation. This paper presented the applicability and universality of the standard forward column/row recurrence equation based on the isolated singular factor method and extended-range arithmetic. Isolating a singular factor is a special normalization method that can improve the universality of the standard forward row recurrence equation to a certain extent, its universality can up to degree hundreds. However, it is invalid for standard forward column recurrence equation. The extended-range arithmetic expands the double-precision number field to the quad-precision number field. The quad-precision number field can retain more significant digits in the operation process and express larger and smaller numbers. The extended-range arithmetic can significantly improve the applicability and universality of the standard forward column/row recurrence equations, its universality can up to degree several thousand. However, the quad-precision number field operation needs to occupy more storage space, which is why its operation speed is slow and undesirable in practical applications. In this paper, the X-number method is introduced in the standard forward row recurrence equation for the first time. With the use of the X-number method, fnALFs can be recursed to 4.2 billion degree by using standard forward column/row recurrence equations.