Numerically satisfactory solutions of Kummer recurrence relations

Pairs of numerically satisfactory solutions as $n\rightarrow \infty$ for the three-term recurrence relations satisfied by the families of functions _1\mbox{F}_1(a+\epsilon_1 n; b +\epsilon_2 n;z), $\epsilon_i \in {\mathbb Z}$, are given. It is proved that minimal solutions always exist, except when $\epsilon_2=0$ and $z$ is in the positive or negative real axis, and that _1\mbox{F}_1 (a+\epsilon_1 n; b +\epsilon_2 n;z) is minimal as $n\rightarrow +\infty$ whenever $\epsilon_2 >0$. The minimal solution is identified for any recurrence direction, that is, for any integer values of $\epsilon_1$ and $\epsilon_2$. When $\epsilon_2\neq 0$ the confluent limit \lim_{b\rightarrow \infty}{}_1\mbox{F}_1(\gamma b;b;z)=e^{\gamma z}, with $\gamma\in{\mathbb C}$ fixed, is the main tool for identifying minimal solutions together with a connection formula; for $\epsilon_2=0$, \lim_{a\rightarrow +\infty} {}_1\mbox{F}_1(a;b;z) /{}_0\mbox{F}_1(;b;az)=e^{z/2} is the main tool to be considered

Numerically satisfactory solutions of hypergeometric recursions

Each family of Gauss hypergeometric functions $$f_n={}_2F_1(a+\varepsilon_1n, b+\varepsilon_2n ;c+\varepsilon_3n; z),\quad n\in {\mathbb Z}\,,$$ for fixed $\varepsilon_j=0,\pm1$ (not all $\varepsilon_j$ equal to zero) satisfies a second order linear difference equation of the form $$A_nf_{n-1}+B_nf_n+C_nf_{n+1}=0.$$ Because of symmetry relations and functional relations for the Gauss functions, many of the 26 cases (for different $\varepsilon_j$ values) can be transformed into each other. In this way, only with four basic difference equations can all other cases be obtained. For each of these recurrences, we give pairs of numerically satisfactory solutions in the regions in the complex plane where $|t_1|\neq |t_2|$, $t_1$ and $t_2$ being the roots of the characteristic equation

Numerical aspects of special functions

This paper describes methods that are important for the numerical evaluation of certain functions that frequently occur in applied mathematics, physics and mathematical statistics. This includes what we consider to be the basic methods, such as recurrence relations, series expansions (both convergent and asymptotic), and numerical quadrature. Several other methods are available and some of these will be discussed in less detail. Examples will be given on the use of special functions in certain problems from mathematical physics and mathematical statistics (integrals and series with special functions)

Numerical aspects of special functions

This paper describes methods that are important for the numerical evaluation of certain functions that frequently occur in applied mathematics, physics and mathematical statistics. This includes what we consider to be the basic methods, such as recurrence relations, series expansions (both convergent and asymptotic), and numerical quadrature. Several other methods are available and some of these will be discussed in less detail. Examples will be given on the use of special functions in certain problems from mathematical physics and mathematical statistics (integrals and series with special functions)

Transitory minimal solutions of hypergeometric recursions and pseudoconvergence of associated continued fractions

Three term recurrence relations yn+1 +bnyn +anyn?1 = 0 can be used for computing recursively a great number of special functions. Depending on the asymptotic nature of the function to be computed, different recursion directions need to be considered: backward for minimal solutions and forward for dominant solutions. However, some solutions interchange their role for finite values of n with respect to their asymptotic behaviour and certain dominant solutions may transitorily behave as minimal. This phenomenon, related to Gautschi’s anomalous convergence of the continued fraction for ratios of confluent hypergeometric functions, is shown to be a general situation which takes place for recurrences with an negative and bn changing sign once. We analyze the anomalous convergence of the associated continued fractions for a number of different recurrence relations (modified Bessel functions, confluent and Gauss hypergeometric functions) and discuss the implication of such transitory behaviour on the numerical stability of recursion