scholarly journals The additivity of polygamma functions

Filomat ◽  
2015 ◽  
Vol 29 (5) ◽  
pp. 1063-1066 ◽  
Author(s):  
Bai-Ni Guo ◽  
Feng Qi ◽  
Qiu-Ming Luo
Keyword(s):  
Unique Root ◽  
Polygamma Functions

In the paper, the authors prove that the functions |?(i)(ex)| for i ? N are subadditive on (ln?i,?) and superadditive on (-?, ln ?i), where ?i ? (0,1) is the unique root of equation 2|?(i)(?)|= ?(i)(?2)|.

scholarly journals Asymptotic expansions of Lambert series and related q-series

2017 ◽  
Vol 13 (08) ◽  
pp. 2097-2113 ◽  
Author(s):  
Shubho Banerjee ◽  
Blake Wilkerson
Keyword(s):  
Asymptotic Expansion ◽  
Asymptotic Expansions ◽  
Theta Functions ◽  
Lambert Series ◽  
Pochhammer Symbol ◽  
Complete Asymptotic Expansion ◽  
Polygamma Functions ◽  
Jacobi Theta Functions ◽  
Relative Errors ◽  
Asymptotic Forms

We study the Lambert series [Formula: see text], for all [Formula: see text]. We obtain the complete asymptotic expansion of [Formula: see text] near [Formula: see text]. Our analysis of the Lambert series yields the asymptotic forms for several related [Formula: see text]-series: the [Formula: see text]-gamma and [Formula: see text]-polygamma functions, the [Formula: see text]-Pochhammer symbol and the Jacobi theta functions. Some typical results include [Formula: see text] and [Formula: see text], with relative errors of order [Formula: see text] and [Formula: see text] respectively.

scholarly journals Accurate approximations to the polygamma functions

1979 ◽  
Vol 37 (2) ◽  
pp. 203-207
Author(s):  
C. Stuart Kelley
Keyword(s):  
Polygamma Functions

scholarly journals Some Inequalities for theq-Analogue of the Classical Riemann Zeta Functions and theq-Polygamma Functions

2014 ◽  
Vol 2014 ◽  
pp. 1-4
Author(s):  
Banyat Sroysang
Keyword(s):  
Zeta Functions ◽  
Polygamma Functions ◽  
Riemann Zeta

We present the generalizations on some inequalities for theq-analogue of the classical Riemann zeta functions and theq-polygamma functions.

scholarly journals Harmonic Numbers and Cubed Binomial Coefficients

2011 ◽  
Vol 2011 ◽  
pp. 1-14
Author(s):  
Anthony Sofo
Keyword(s):  
Closed Form ◽  
Binomial Coefficients ◽  
Harmonic Numbers ◽  
Polygamma Functions ◽  
Form Representation ◽  
The Given

Euler related results on the sum of the ratio of harmonic numbers and cubed binomial coefficients are investigated in this paper. Integral and closed-form representation of sums are developed in terms of zeta and polygamma functions. The given representations are new.

scholarly journals COMPLETE MONOTONICITY OF A FUNCTION INVOLVING THE DIVIDED DIFFERENCE OF PSI FUNCTIONS

2013 ◽  
Vol 88 (2) ◽  
pp. 309-319 ◽  
Author(s):  
FENG QI ◽  
PIETRO CERONE ◽  
SEVER S. DRAGOMIR
Keyword(s):  
Divided Difference ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Complete Monotonicity ◽  
Divided Differences ◽  
Psi Function ◽  
Gamma Functions ◽  
Completely Monotonic ◽  
Polygamma Functions ◽  
Necessary And Sufficient

AbstractNecessary and sufficient conditions are presented for a function involving the divided difference of the psi function to be completely monotonic and for a function involving the ratio of two gamma functions to be logarithmically completely monotonic. From these, some double inequalities are derived for bounding polygamma functions, divided differences of polygamma functions, and the ratio of two gamma functions.

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