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We present new functional inequalities and monotonicity results for the modified Lommel functions of the first kind, such as absolute monotonicity, complete monotonicity, monotonicity, log-convexity, and strong convexity. Various inequalities involving reverse Turán-type inequalities are proven to illustrate the results of this paper. [ABSTRACT FROM AUTHOR]

Bertoin, Roynette and Yor (2004) described new connections between the class LE of Lévy‐Laplace exponents Ψ (also called the class of (sub)critical branching mechanism) and the class of Bernstein functions (BF) which are internal, that is, those Bernstein functions ϕ s.t. Ψ∘ϕ remains a Bernstein function for every Ψ. We complete their work and illustrate how the class of internal function is rich from the stochastic point of view. It is well known that every ϕ∈BF corresponds to (i) a subordinator (Xt)t ≥ 0 (or equivalently to transition semigroups ℙ(Xt∈dx)t≥0 and (ii) a Lévy measure μ (which controls the jumps of the subordinator). It is also known that, on (0,∞), the measure ℙ(Xt∈dx)/t converges vaguely to dδ0(dx)+μ(dx) as t→0, where d is the drift term, but rare are the situations where we can compare the transition semigroups with the Lévy measure. Our extensive investigations on the composition of Lévy‐Laplace exponents Ψ with Bernstein functions show, for instance, these remarkable facts: ϕ is internal is equivalent to (a) ϕ2∈BF or to (b) tμ(dx)−ℙ(Xt∈dx) is a positive measure on (0,∞). We also provide conditions on μ insuring internality for ϕ and illustrate how Lévy‐Laplace exponents are closely connected to the class of Thorin Bernstein function. [ABSTRACT FROM AUTHOR]

In this paper, a generic fractional derivative is defined as a set of the linear operators left-inverse to the Riemann–Liouville fractional integral. Then, the theory of the left-invertible operators developed by Przeworska-Rolewicz is applied to deduce its properties. In particular, we characterize its domain, null-space, and projector operator; establish the interrelations between its different realizations; and present a generalized fractional Taylor formula involving the generic fractional derivative. Then, we consider the fractional relaxation equation containing the generic fractional derivative, derive a closed-form formula for its unique solution, and study its complete monotonicity. [ABSTRACT FROM AUTHOR]

In this study, using convolution theorem of the Laplace transforms, a monotonicity rule for the ratio of two Laplace transforms, Bernstein’s theorem for completely monotonic functions, and other analytic techniques, the authors verify decreasing property of a ratio between three derivatives of a function involving trigamma function and find the necessary and sufficient conditions for a function defined by three derivatives of a function involving trigamma function to be completely monotonic. These results confirm previous guesses posed by Qi and generalize the corresponding known conclusions.

In this paper, we find some conditions under which a sequence \{\alpha (n)\} will satisfy the Laguerre inequality of any order asymptotically. Using this method, we prove that for any r and some constant c, the Maclaurin coefficients \gamma (n) of the Riemann Xi-function satisfy the Laguerre inequality of order r when n>cr^3, which provides a necessary condition for the Riemann hypothesis. We also prove that the partition function satisfies the Laguerre inequality of order r\geq 5 when n\geq 6r^4. As a consequence, it gives an affirmative answer to Wagner's conjecture on the threshold for the Laguerre inequalities of order no more than 10 for the partition function. Moreover, motivated by the study of Craven and Csordas on the complete monotonicity of the Maclaurin coefficients of entire functions in Laguerre-Pólya class, we consider the complete monotonicity of the sequences \{\alpha (n)\}. We give the criteria for the asymptotically complete monotonicity of the sequence \{\alpha (n)\} and \{\log \alpha (n)\}, respectively. With this criteria, we show that (-1)^r \Delta ^r \gamma (n)>0 for n>ce^{r^3} and (-1)^{r-1} \Delta ^r \log \gamma (n)>0 for n>cr^2. Furthermore, we propose some open problems. [ABSTRACT FROM AUTHOR]

Sparse structure learning in high‐dimensional Gaussian graphical models is an important problem in multivariate statistical inference, since the sparsity pattern naturally encodes the conditional independence relationship among variables. However, maximum a posteriori (MAP) estimation is challenging under hierarchical prior models, and traditional numerical optimization routines or expectation–maximization algorithms are difficult to implement. To this end, our contribution is a novel local linear approximation scheme that circumvents this issue using a very simple computational algorithm. Most importantly, the condition under which our algorithm is guaranteed to converge to the MAP estimate is explicitly stated and is shown to cover a broad class of completely monotone priors, including the graphical horseshoe. Further, the resulting MAP estimate is shown to be sparse and consistent in the ℓ2$$ {\ell}_2 $$‐norm. Numerical results validate the speed, scalability and statistical performance of the proposed method. [ABSTRACT FROM AUTHOR]

The principal tool of computation used in this paper is classical Gaussian quadrature on the interval [0,1], which happens to be particularly effective here. Explicit expressions are found for the derivatives of the Ramanujan integral in question, and it is proved that the latter is completely monotone on (0,\infty). As a byproduct, known series expansions for incomplete gamma functions are examined with regard to their convergence properties. The paper also pays attention to another famous integral, the Euler integral — better known as the gamma function — revitalizing a largely neglected part of the function, the part corresponding to negative values of the argument, which plays a prominent role in our work. [ABSTRACT FROM AUTHOR]

The Wallis power function is defined on \left (-\min \left \{ p,q\right \},\infty \right) by \begin{equation*} W_{p,q}\left (x\right) =\left (\dfrac {\Gamma \left (x+p\right) }{\Gamma \left (x+q\right) }\right) ^{1/\left (p-q\right) }\text { if }p\neq q\text { and }W_{p,p}\left (x\right) =e^{\psi \left (x+p\right) }. \end{equation*} Let p_{i},q_{i}\in \mathbb {R} with 0<\delta _{i}=p_{i}-q_{i}\leq 1, \theta _{i}=\left (1-\delta _{i}\right) /2 for i=1,2 and p_{1}+q_{1}=p_{2}+q_{2}=2\sigma +1. We prove that, if q_{1}>q_{2} then for any integer m\in \mathbb {N}, the function \begin{equation*} x\mapsto \left (-1\right) ^{m}\left [ \ln \frac {W_{p_{1},q_{1}}\left (x\right) }{W_{p_{2},q_{2}}\left (x\right) }-\sum _{k=1}^{m}\frac {a_{2k}\left (\theta _{1},\theta _{2}\right) }{\left (2k+1\right) \left (2k\right) \left (x+\sigma \right) ^{2k}}\right ] \end{equation*} is completely monotonic on \left (-\sigma,\infty \right), where \begin{equation*} a_{2k}\left (\theta _{1},\theta _{2}\right) =\frac {B_{2k+1}\left (\theta _{2}\right) }{\theta _{2}-1/2}-\frac {B_{2k+1}\left (\theta _{1}\right) }{\theta _{1}-1/2}. \end{equation*} This extends and generalizes some known results. [ABSTRACT FROM AUTHOR]

Let p, q ∈ ℝ with p − q ≽ 0, σ = 1 2 (p + q − 1) and s = 1 2 (1 − p + q) , and let D m (x ; p , q) = D 0 (x ; p , q) + ∑ k = 1 m B 2 k (s) 2 k (x + σ) 2 k , where D 0 (x ; p , q) = ψ (x + p) + ψ (x + q) 2 − ln (x + σ). We establish the asymptotic expansion D 0 (x ; p , q) ∼ − ∑ n = 1 ∞ B 2 n (s) 2 n (x + σ) 2 n a s x → ∞ , where B2n(s) stands for the Bernoulli polynomials. Further, we prove that the functions (− 1) m D m (x ; p , q) and (− 1) m + 1 D m (x ; p , q) are completely monotonic in x on (−σ, ∞) for every m ∈ ℕ0 if and only if p − q ∈ [ 0 , 1 2 ] and p − q = 1, respectively. This not only unifies the two known results but also yields some new results. [ABSTRACT FROM AUTHOR]

Using the differential operator Ω s : = s I - x d d x , s > 0 , we build a new class of infinitely divisible distributions on the half-line. For this class, we give a stochastic interpretation and we provide several monotonicity properties for the associated subordinators. As an application, we solve a problem raised separately by Sendov and Shan in (J Theor Probab 28:1689–1725, 2015) and by Simon in (Math Nachr 285(4): 497–506, 2012) on the distribution of the stable subordinators. Finally, we provide a new complete monotonicity property for the Mittag-Leffler function. [ABSTRACT FROM AUTHOR]

The article provides construction algorithms for consistent model classes of continuous spherical and elliptical distributions. The algorithms are based on characterization theorems for consistent families of generator functions. This characterization uses the term of complete monotonicity. The algorithms are applied to several examples of generators leading to consistent families of generators with explicit formulas for marginal densities of arbitrary dimension.

In this paper, a generic fractional derivative is defined as a set of the linear operators left-inverse to the Riemann–Liouville fractional integral. Then, the theory of the left-invertible operators developed by Przeworska-Rolewicz is applied to deduce its properties. In particular, we characterize its domain, null-space, and projector operator; establish the interrelations between its different realizations; and present a generalized fractional Taylor formula involving the generic fractional derivative. Then, we consider the fractional relaxation equation containing the generic fractional derivative, derive a closed-form formula for its unique solution, and study its complete monotonicity.

In this paper, the authors review and survey some results published since 2020 about (complete) monotonicity, inequalities, and their necessary and sufficient conditions for several newly introduced functions involving polygamma functions and originating from the estimation of the sectional curvature of the Fisher–Rao geometry of the Dirichlet distributions in the two-dimensional case. [ABSTRACT FROM AUTHOR]

Let a,b,c,d ∈ R with a+b = c+d = 2r +1. Then In Γ(x+a) Γ (x+b)/Γ(x+c) Γ (x+d) ~ ∞/Σ/k=1 B2k (θ1) - B2k (θ2)/k (2k-1) (x+r)2k-1 as x→∞, where (δ1,δ2) = (|a−b|,|c−d|) = (1−2θ1,1−2θ2). When 0 ≤ δ2 < δ1 ≤ 1, the function x→ (-1)m [1n Γ(x+a) Γ (x+b)/Γ(x+c) Γ (x+d) - m/Σ/k=1 B2k (θ1) - B2k (θ2)/k (2k-1) (x+r)2k-1 ] for m ∈ N is completely monotonic on (−r,∞). This yields some known and new results. [ABSTRACT FROM AUTHOR]

We analyze the role played by functions with positive differences defined on convex cones. In particular, we study functions that satisfy linear functional inequalities that extend the three-variable Hornich-Hlawka functional inequality, f x + f y + f z + f x + y + z ≥ f x + y + f y + z + f z + x + f (0) , especially to the case of n variables. Beyond the classical setting, we present extensions to the case of positive operators. [ABSTRACT FROM AUTHOR]

We theoretically analyze the problem of testing for p‐hacking based on distributions of p‐values across multiple studies. We provide general results for when such distributions have testable restrictions (are non‐increasing) under the null of no p‐hacking. We find novel additional testable restrictions for p‐values based on t‐tests. Specifically, the shape of the power functions results in both complete monotonicity as well as bounds on the distribution of p‐values. These testable restrictions result in more powerful tests for the null hypothesis of no p‐hacking. When there is also publication bias, our tests are joint tests for p‐hacking and publication bias. A reanalysis of two prominent data sets shows the usefulness of our new tests.

In this article, with the help of an integral representation of the Dirichlet lambda function, by means of a monotonicity rule for the ratio of two integrals with a parameter, and by virtue of complete monotonicity and another property of an elementary function involving the exponential function, the authors find increasing property and logarithmic convexity of two functions containing the gamma function and the Dirichlet lambda function.

Hyperbolic complete monotonicity property (HCM) is a way to check if a distribution is a generalized gamma (GGC), hence is infinitely divisible. In this work, we illustrate to which extent the Mittag-Leffler functions  E α , α ∈ (0 , 2 ] , enjoy the  HCM property, and then intervene deeply in the probabilistic context. We prove that for suitable  α and complex numbers z, the real and imaginary part of the functions  x ↦ E α z x , are tightly linked to the stable distributions and to the generalized Cauchy kernel. [ABSTRACT FROM AUTHOR]

The theory of time scale was proposed to unite continuous and discrete statements. In this paper, we introduce the concepts of complete monotonicity, logarithmic complete monotonicity, and absolute monotonicity of multivariate functions on time scales utilizing the delta derivative. Following that, we investigate the properties of delta complete monotonicity on time scales and present some judgment rules. Using judgment rules, three functions are shown to be delta complete monotonicity on time scales. And then, in order to quantitatively measure two delta completely monotonic functions on time scales, we present the concept of completely monotonic degree of univariate and multivariate functions on time scales and explore some properties of them. [ABSTRACT FROM AUTHOR]

A combinatorial proof of the Gaussian product inequality (GPI) is given under the assumption that each component of a centered Gaussian random vector X=(X1,…,Xd){\boldsymbol{X}}=\left({X}_{1},\ldots ,{X}_{d}) of arbitrary length can be written as a linear combination, with coefficients of identical sign, of the components of a standard Gaussian random vector. This condition on X{\boldsymbol{X}} is shown to be strictly weaker than the assumption that the density of the random vector (∣X1∣,…,∣Xd∣)\left(| {X}_{1}| ,\ldots ,| {X}_{d}| ) is multivariate totally positive of order 2, abbreviated MTP2{\text{MTP}}_{2}, for which the GPI is already known to hold. Under this condition, the paper highlights a new link between the GPI and the monotonicity of a certain ratio of gamma functions.

For r, s ∈ R and ρ = min {r, s}, let D [x + r, x + s; ψn-1] ≡ -φn (x) be the divided difference of the functions ψn-1 = (-1)n ψ(n-1) (n ∈ N) on (-ρ, ∞), where ψ(n) stands for the polygamma functions. In this paper, we present the necessary and sufficient conditions for the functions... to be completely monotonic on (-ρ, ∞), where mi, ni ∈ N for i = 1, ..., k with k ≥ 2 and .... These generalize known results and gives an answer to a problem. [ABSTRACT FROM AUTHOR]

We establish some identities in law for the convolution of a beta prime distribution with itself, involving the square root of beta distributions. The proof of these identities relies on transformations on generalized hypergeometric series obtained via Appell series of the first kind and Thomae's relationships for {}_3F_2(1). Using a self-decomposability argument, the identities are applied to derive complete monotonicity properties for quotients of confluent hypergeometric functions having a doubling character. By means of probability, we also obtain a simple proof of Turán's inequality for the parabolic cylinder function and the confluent hypergeometric function of the second kind. The case of Mill's ratio is discussed in detail. [ABSTRACT FROM AUTHOR]

In this paper, the authors review and survey some results published since 2020 about (complete) monotonicity, inequalities, and their necessary and sufficient conditions for several newly introduced functions involving polygamma functions and originating from the estimation of the sectional curvature of the Fisher–Rao geometry of the Dirichlet distributions in the two-dimensional case.

Hyperbolic complete monotonicity property (HCM) is a way to check if a distribution is a generalized gamma (GGC), hence is infinitely divisible. In this work, we illustrate to which extent the Mittag-Leffler functions Eα,α∈(0,2], enjoy the HCM property, and then intervene deeply in the probabilistic context. We prove that for suitable α and complex numbers z, the real and imaginary part of the functions x↦Eαzx, are tightly linked to the stable distributions and to the generalized Cauchy kernel.

Abstract Let Γ ( x ) $\Gamma (x)$ denote the classical Euler gamma function. We set ψ n ( x ) = ( − 1 ) n − 1 ψ ( n ) ( x ) $\psi _{n}(x)=(-1)^{n-1}\psi ^{(n)}(x)$ ( n ∈ N $n\in \mathbb{N}$ ), where ψ ( n ) ( x ) $\psi ^{(n)}(x)$ denotes the nth derivative of the psi function ψ ( x ) = Γ ′ ( x ) / Γ ( x ) $\psi (x)=\Gamma '(x)/\Gamma (x)$ . For λ, α, β ∈ R $\beta \in \mathbb{R}$ and m , n ∈ N $m,n\in \mathbb{N}$ , we establish necessary and sufficient conditions for the functions L ( x ; λ , α , β ) = ψ m + n ( x ) − λ ψ m ( x + α ) ψ n ( x + β ) $$ L(x;\lambda ,\alpha ,\beta )=\psi _{m+n}(x)-\lambda \psi _{m}(x+ \alpha ) \psi _{n}(x+\beta ) $$ and − L ( x ; λ , α , β ) $-L(x;\lambda ,\alpha ,\beta )$ to be completely monotonic on ( − min ( α , β , 0 ) , ∞ ) $(-\min (\alpha ,\beta ,0),\infty )$ . As a result, we generalize and refine some inequalities involving the polygamma functions and also give some inequalities in terms of the ratio of gamma functions.

With the aid of convolution theorem of the Laplace transforms, a monotonicity rule for the ratio of two Laplace transforms, Bernstein's theorem for completely monotonic functions, and other analytic techniques, the author presents decreasing property of a ratio constituted via three derivatives of a sum involving trigamma function and discovers necessary and sufficient conditions for a function constituted via three derivatives of a function involving trigamma function to be completely monotonic. [ABSTRACT FROM AUTHOR]

In this work we prove that some integrals of special functions are positive by applying the Plancherel theorem for Hankel transforms and positivity of the modified Bessel functions. We also prove that, except an extra elementary factor, Hankel transforms map subsets of completely monotonic functions into complete monotonic functions. [ABSTRACT FROM AUTHOR]

Let K_{\nu } be the modified Bessel functions of the second kind of order \nu. The ratio Q_{\nu }\left (x\right) =xK_{\nu -1}\left (x\right) /K_{\nu }\left (x\right) appeared in physics and probability. In this paper, we collate properties of this ratio, and prove the conjecture that \left (-1\right) ^{n}Q_{\nu }^{\left (n\right) }\left (x\right) >\left (<\right) 0 for x>0 and n=2,3 if \left \vert \nu \right \vert >\left (<\right) 1/2 holds for n=2. This yields several new consequences and improves some known results. Finally, two conjectures are proposed. [ABSTRACT FROM AUTHOR]

In the paper, the authors introduce a matrix-parametrized generalization of the multinomial probability mass function that involves a ratio of several multivariate gamma functions. They show the logarithmically complete monotonicity of this generalization and derive several inequalities involving ratios of multivariate gamma functions. [ABSTRACT FROM AUTHOR]

We give some necessary and some sufficient conditions for the complete monotonicity on the negative half-line of a Mittag–Leffler function of Le Roy type. It is conjectured that the underlying positive random variable, when it exists, must be logarithmically infinitely divisible. [ABSTRACT FROM AUTHOR]

In this work, we refine the results of Sendov and Shan [New representation theorems for completely monotone and Bernstein functions with convexity properties on their measures, J. Theor. Probab. 28 (2015), 1689–1725] on subordinators obtained by the class of Bernstein functions stable by the Mellin-Euler differential operator I−xddxI-x\tfrac{\text{d}}{\text{d}x} by giving a stochastic interpretation, proving monotonicity properties of the related distributions and providing additional extensions.

Let Γ (x) denote the classical Euler gamma function. We set ψ n (x) = (− 1) n − 1 ψ (n) (x) (n ∈ N ), where ψ (n) (x) denotes the nth derivative of the psi function ψ (x) = Γ ′ (x) / Γ (x) . For λ, α, β ∈ R and m , n ∈ N , we establish necessary and sufficient conditions for the functions L (x ; λ , α , β) = ψ m + n (x) − λ ψ m (x + α) ψ n (x + β) and − L (x ; λ , α , β) to be completely monotonic on (− min (α , β , 0) , ∞) . As a result, we generalize and refine some inequalities involving the polygamma functions and also give some inequalities in terms of the ratio of gamma functions. [ABSTRACT FROM AUTHOR]

Time-fractional initial-boundary problems of parabolic type are considered. Previously, global error bounds for computed numerical solutions to such problems have been provided by Liao et al. (SIAM J. Numer. Anal. 2018, 2019) and Stynes et al. (SIAM J. Numer. Anal. 2017). In the present work we show how the concept of complete monotonicity can be combined with these older analyses to derive local error bounds (i.e., error bounds that are sharper than global bounds when one is not close to the initial time t = 0 {t=0}). Furthermore, we show that the error analyses of the above papers are essentially the same – their key stability parameters, which seem superficially different from each other, become identical after a simple rescaling. Our new approach is used to bound the global and local errors in the numerical solution of a multi-term time-fractional diffusion equation, using the L1 scheme for the temporal discretisation of each fractional derivative. These error bounds are α-robust. Numerical results show they are sharp. [ABSTRACT FROM AUTHOR]

In the paper, by virtue of convolution theorem for the Laplace transforms and analytic techniques, the author finds necessary and sufficient conditions for complete monotonicity, monotonicity, and inequalities of several functions involving polygamma functions. By these results, the author derives a lower bound of a function related to the sectional curvature of the Fisher–Rao manifold of beta distributions. [ABSTRACT FROM AUTHOR]

We study the preservation of impulse response non-negativity, also known as external positivity, for general discretization methods. The results yield new subsets in the space of transfer function coefficients which ensure external positivity. Consequently, linear programming approaches for designing controllers ensuring monotonic step responses are derived for both continuous and discrete-time systems. [ABSTRACT FROM AUTHOR]

Abstract Several conjectures posed by Qi on completely monotonic degrees of remainders for the asymptotic formulas of the digamma and trigamma functions are proved.

Abstract In this paper, we prove new complete monotonicity properties of some functions related to generalized k-gamma and k-polygamma functions. Applications of the results yield various new inequalities. In the end, double inequalities are constructed involving the k-generalized digamma and polygamma functions.

In the paper, by virtue of convolution theorem for the Laplace transforms, Bernstein's theorem for completely monotonic functions, some properties of a function involving exponential function, and other analytic techniques, the author finds necessary and sufficient conditions for two functions defined by two derivatives of a function involving trigamma function to be completely monotonic or monotonic. These results generalize corresponding known ones. [ABSTRACT FROM AUTHOR]

The aim of this work is to derive several analytical properties for the class of entire functions defined by ℐ ϕ (t) = ∑ n = 0 ∞ 1 W ϕ (n + 1) t n n ! with W ϕ (1) = 1 , W ϕ (n + 1) = ∏ k = 1 n ϕ (k) , where the parameter ϕ ranges over the set of Bernstein functions. This includes the expression of their order and type, given explicitly in terms of the so-called Blumenthal-Getoor lower index, the derivation of Turán-type inequalities and several complete monotonicity properties. This unifies the proof and provides new results for a wide range of special functions since this class encompasses, as special instances, the modified Bessel functions, hypergeometric functions, the Mittag-Leffler functions, the Wright functions and q-series. [ABSTRACT FROM AUTHOR]

In this paper, a new set of sufficient conditions are imposed on the parameters of the Fox H-functions, which allow us to conclude that it is non-negative. As applications, some classes of completely monotonic and positive definite functions involving the Fox H-function are established. Some new facts regarding the Fox-Wright functions, including complete monotonicity, logarithmic completely monotonicity and monotonicity of their ratios are discussed. [ABSTRACT FROM AUTHOR]

In this paper we establish some Turan type inequalities for the τ-hypergeometric functions. We further prove the monotonicity of ratios of the τ-hypergeometric functions and it is depicted how our results are closely connected with certain Turan type inequalities. Additionally, we prove some results which correspond to the sub-additivity properties of the t-Gauss hypergeometric functions. We also present an open problem which may be of interest for further investigation. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we prove complete monotonicity of some functions involving k-polygamma functions. As an application of the main result, we also give new upper and lower bounds of the k-digamma function.

Abstract In the paper, we solve one conjecture on an inequality involving digamma function, an open problem, and a conjecture on monotonicity of functions involving generalized digamma function. We also prove a new inequality for digamma function.

By using some analytical techniques, we prove a complete monotonicity property of a certain function involving the (p, k)-digamma function. Subsequently, we derive some inequalities for the (p, k)- digamma function. As special cases of the established results, we deduce some new results concerning the p-digamma and the k-digamma functions. Our results are extensions of some previous results due to Qiu and Vuorinen, Mortici, and Merovci.

Using the Bernstein theorem we prove the complete monotonicity of the three parameter Mittag-Leffler function Eα, β (-w) for w > 0 and suitably constrained parameters α, β and γ. [ABSTRACT FROM AUTHOR]

In this paper our aim is to establish new integral representations for the Fox–Wright function when In particular, closed-form integral expressions are derived for the four parameter Wright function under a special restriction on parameters. Exponential bounding inequalities are derived for a class of the Fox–Wright function. Moreover, complete monotonicity property is presented for these functions. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we establish a concave theorem and some inequalities for the generalized digamma function. Hence, we give complete monotonicity property of a determinant function involving all kinds of derivatives of the generalized digamma function.