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For real values of the argument  and nonnegative integer  , the differentiated gamma functions  ,  ,  , and  are real (or infinity). The function  is real (or infinity) for real values of argument  and integer  .
The differentiated gamma functions  ,  ,  , and  have simple values for zero arguments:
The differentiated gamma functions  ,  ,  , and  with rational arguments can sometimes be evaluated through classical constants and logarithms, for example:
The previous relations are particular cases of the following general formulas:
The differentiated gamma functions  and  with integer parameters  and  have the following representations:
The digamma function  and the harmonic number  are defined for all complex values of the variable  . The functions  and  are analytical functions of  and  over the whole complex  ‐ and  ‐planes. For fixed  , the generalized harmonic number  is an entire function of  .
The differentiated gamma functions  and  have an infinite set of singular points  , where  for  and  for  . These points are the simple poles with residues  . The point  is the accumulation point of poles for the functions  ,  , and  (with fixed nonnegative integer  ), which means that  is an essential singular point.
For fixed nonnegative integer  , the function  has an infinite set of singular points:  are the simple poles with residues  ; and  are the poles of order  with residues  .
For fixed  , the function  does not have poles and the function  has only one singular point at  , which is an essential singular point.
The functions  and  do not have branch points and branch cuts.
For integer  , the function  does not have branch points and branch cuts.
For fixed noninteger  , the function  has two singular branch points  and  , and it is a single‐valued function on the  ‐plane cut along the interval  , where it is continuous from above:
For fixed  , the functions  and  do not have branch points and branch cuts.
The differentiated gamma functions  ,  ,  , and  do not have periodicity.
The differentiated gamma functions  ,  ,  , and  have mirror symmetry:
The differentiated gamma functions  ,  , and  have the following series expansions near regular points:
Near singular points, the differentiated gamma functions  ,  ,  , and  can be expanded through the following series:
Here  is the Euler gamma constant  .
Except for the generalized power series, there are other types of series through which differentiated gamma functions  ,  ,  , and  can be represented, for example:
The asymptotic behavior of the differentiated gamma functions  ,  ,  , and  can be described by the following formulas (only the main terms of the asymptotic expansions are given):
The differentiated gamma functions  ,  ,  , and  can also be represented through the following equivalent integrals:
The following formulas describe some of the transformations that change the differentiated gamma functions into themselves:
Transformations with arguments that are integer multiples take the following forms:
The following transformations represent summation theorems:
The differentiated gamma functions  ,  ,  , and  satisfy the following recurrence identities:
The previous formulas can be generalized to the following recurrence identities with a jump of length  :
The derivatives of the differentiated gamma functions  ,  ,  , and  have rather simple representations:
The corresponding symbolic  -order derivatives of all the differentiated gamma functions  ,  ,  , and  can be expressed by the following formulas:
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