Evaluations of series of the qq-Watson, qq-Dixon, and qq-Whipple type

Using $q$-series identities and series rearrangement, we establish several extensions of $q$-Watson formulas with two extra integer parameters. Then they and Sears' transformation formula are utilized to derive some generalizations of $q$-Dixon formulas and $q$-Whipple formulas with two extra integer parameters. As special cases of these results, many interesting evaluations of series of $q$-Watson,$q$-Dixon, and $q$-Whipple type are displayed

Algorithms for the indefinite and definite summation

The celebrated Zeilberger algorithm which finds holonomic recurrence equations for definite sums of hypergeometric terms $F(n,k)$ is extended to certain nonhypergeometric terms. An expression $F(n,k)$ is called a hypergeometric term if both $F(n+1,k)/F(n,k)$ and $F(n,k+1)/F(n,k)$ are rational functions. Typical examples are ratios of products of exponentials, factorials, $\Gamma$ function terms, bin omial coefficients, and Pochhammer symbols that are integer-linear with respect to $n$ and $k$ in their arguments. We consider the more general case of ratios of products of exponentials, factorials, $\Gamma$ function terms, binomial coefficients, and Pochhammer symbols that are rational-linear with respect to $n$ and $k$ in their arguments, and present an extended version of Zeilberger's algorithm for this case, using an extended version of Gosper's algorithm for indefinite summation. In a similar way the Wilf-Zeilberger method of rational function certification of integer-linear hypergeometric identities is extended to rational-linear hypergeometric identities

Summation formulas for Fox-Wright function

By means of inversion techniques and several known hypergeometric series identities, summation formulas for Fox-Wright function are explored. They give some new hypergeometric series identities when the parameters are specified

Identities for the gamma and hypergeometric functions: an overview from Euler to the present

A research report submitted to the Faculty of Science, University of the Witwatersrand, in fulfilment of the requirements for the degree of Master of Science. Johannesburg, 2013Equations involving the gamma and hypergeometric functions are of great interest to mathematicians and scientists, and newly proven identities for these functions assist in finding solutions to differential and integral equations. In this work we trace a brief history of the development of the gamma and hypergeometric functions, illustrate the close relationship between them and present a range of their most useful properties and identities, from the earliest ones to those developed in more recent years. Our literature review will show that while continued research into hypergeometric identities has generated many new results, some of these can be shown to be variations of known identities. Hence, we will also discuss computer based methods that have been developed for creating and analysing such identities, in order to check for originality and for numerical validity

Parental risk and resilience: How does evidence inform child maltreatment prevention and reduction?

Child Maltreatment is a global concern with a sequela of negative consequences. The Risk and Resilience Ecological Framework is used to enable synthesis of evidence from two systematic reviews, A and B, on evidence of factors that influence parental child maltreatment. Review A comprises non-interventional, empirical studies to determine parental risk and protective factor interplay, lending support to causal and correlational links to child maltreatment. Review B synthesises evidence from intervention evaluations on parental risk factors and intervention provision for child maltreatment. A total of 128 studies, 68 observational studies in Review A and 60 intervention evaluations in Review B, were systematically reviewed. Quality appraisal did not lead to exclusion of studies. Review A findings mirror prior evidence and highlight nuances such as memories of parental childhood maltreatment as risk, emotional support for mothers and companionship support for fathers as protective, and demarcate maltreatment type-specific factors, especially for physical abuse and neglect. A low representation of fathers, under-research of unique factors for sexual and emotional abuse and of macro-level protective factors were identified. Review B provides comprehensive data on potentially effective intervention components including child development education and parental emotional regulation. Behaviour Change Techniques Framework helped identify potentially optimal delivery techniques including Instruction on how to perform a behaviour and Social support (unspecified). Lack of cultural representation, sparsity of interventions targeting fathers, over-reliance on self-reporting measures and under-examination of macro-level intervention components were identified as gaps in knowledge. Both reviews underline a call for consensus in definitions and avoidance of umbrella terms. A final synthesis elucidated the complex interplay of multiple influences on parental child maltreatment. Findings offer valuable insight to move the field forward, inform researchers, policy, and practice to strengthen parental resilience to prevent and reduce child maltreatment