where {{mvar|B<sub>k</sub>}} is the {{mvar|k}}th [[Bernoulli number]] and {{mvar|ζ}} is the [[Riemann zeta function]]. Although the infinite sum converges for no {{mvar|x}}, this expansion becomes more accurate for larger values of {{mvar|x}} and '''any finite partial sum''' cut off from the full series. To compute {{math|''ψ''(''x'')}} for small {{mvar|x}}, the recurrence relation
where {{mvar|B<sub>k</sub>}} is the {{mvar|k}}th [[Bernoulli number]] and {{mvar|ζ}} is the [[Riemann zeta function]]. Although the infinite sum does not converge for any {{mvar|x}}, this expansion becomes more accurate for larger values of {{mvar|x}} and '''any finite partial sum''' cut off from the full series. To compute {{math|''ψ''(''x'')}} for small {{mvar|x}}, the recurrence relation
which converges for |z| < 1. Here, ζ(n) is the Riemann zeta function. This series is easily derived from the corresponding Taylor's series for the Hurwitz zeta function.
Newton series
The Newton series for the digamma follows from Euler's integral formula:
Actually, ψ is the only solution of the functional equation
that is monotone on ℝ+ and satisfies F(1) = −γ. This fact follows immediately from the uniqueness of the Γ function given its recurrence equation and convexity restriction. This implies the useful difference equation:
Some finite sums involving the digamma function
There are numerous finite summation formulas for the digamma function. Basic summation formulas, such as
are due to Gauss.[3][4] More complicated formulas, such as
are due to works of certain modern authors (see e.g. Appendix B in Blagouchine (2014)[5]).
Gauss's digamma theorem
For positive integers r and m (r < m), the digamma function may be expressed in terms of Euler's constant and a finite number of elementary functions
which holds, because of its recurrence equation, for all rational arguments.
the digamma function for x, also a real number, can be approximated by
which is the beginning of the asymptotical expansion of ψ(x). The full asymptotic series of this expansions is
where Bk is the kth Bernoulli number and ζ is the Riemann zeta function. Although the infinite sum does not converge for any x, this expansion becomes more accurate for larger values of x and any finite partial sum cut off from the full series. To compute ψ(x) for small x, the recurrence relation
can be used to shift the value of x to a higher value. Beal[7] suggests using the above recurrence to shift x to a value greater than 6 and then applying the above expansion with terms above x14 cut off, which yields "more than enough precision" (at least 12 digits except near the zeroes).
As x goes to infinity, ψ(x) gets arbitrarily close to both ln(x − 1/2) and ln x. Going down from x + 1 to x, ψ decreases by 1 / x, ln(x − 1/2) decreases by ln (x + 1/2) / (x − 1/2), which is more than 1 / x, and ln x decreases by ln (1 + 1 / x), which is less than 1 / x. From this we see that for any positive x greater than 1/2,
or, for any positive x,
The exponential exp ψ(x) is approximately x − 1/2 for large x, but gets closer to x at small x, approaching 0 at x = 0.
For x < 1, we can calculate limits based on the fact that between 1 and 2, ψ(x) ∈ [−γ, 1 − γ], so
or
From the above asymptotic series for ψ, one can derive an asymptotic series for exp(−ψ(x)). The series matches the overall behaviour well, that is, it behaves asymptotically as it should for large arguments, and has a zero of unbounded multiplicity at the origin too.
This can be considered a Taylor expansion of exp(−ψ(1 / y)) at y = 0, but it does not converge.[8]
Another expansion is more precise for large arguments and saves computing terms of even order.
Special values
The digamma function has values in closed form for rational numbers, as a result of Gauss's digamma theorem. Some are listed below:
Moreover, by the series representation, it can easily be deduced that at the imaginary unit,
Roots of the digamma function
The roots of the digamma function are the saddle points of the complex-valued gamma function. Thus they lie all on the real axis. The only one on the positive real axis is the unique minimum of the real-valued gamma function on ℝ+ at x0 = 1.461632144968.... All others occur single between the poles on the negative axis:
holds asymptotically. A better approximation of the location of the roots is given by
and using a further term it becomes still better
which both spring off the reflection formula via
and substituting ψ(xn) by its not convergent asymptotic expansion. The correct second term of this expansion is 1 / 2n, where the given one works good to approximate roots with small n.
Regarding the zeros, the following infinite sum identities were recently proved by István Mező[9]
^If it converged to a function f(y) then ln(f(y) / y) would have the same Maclaurin series as ln(1 / y) − φ(1 / y). But this does not converge because the series given earlier for φ(x) does not converge.
^Mező, István (2014). "A note on the zeros of the Digamma function and the derivative of the log-Barnes function". arXiv:1409.2971v1.