A generic polynomial solution for the differential equation of hypergeometric type and six sequences of orthogonal polynomials related to it

In this work, we present a generic formula for the polynomial solution families of the well-known differential equation of hypergeometric type s(x)y"n(x) + t(x)y'n(x) - lnyn(x) = 0 and show that all the three classical orthogonal polynomial families as well as three finite orthogonal polynomial families, extracted from this equation, can be identified as special cases of this derived polynomial sequence. Some general properties of this sequence are also given.

A generalization of Student’s t-distribution from the viewpoint of special functions

Student’s t-distribution has found various applications in mathematical statistics. One of the main properties of the t-distribution is to converge to the normal distribution as the number of samples tends to infinity. In this paper, by using a Cauchy integral we introduce a generalization of the t-distribution function with four free parameters and show that it converges to the normal distribution again. We provide a comprehensive treatment of mathematical properties of this new distribution. Moreover, since the Fisher F-distribution has a close relationship with the t-distribution, we also introduce a generalization of the F-distribution and prove that it converges to the chi-square distribution as the number of samples tends to infinity. Finally some particular sub-cases of these distributions are considered.

Double-autoionization decay of resonantly excited single-electron states

Perturbation theory in the lowest non-vanishing order in interelectron interaction has been applied to the theoretical investigation of double-ionization decays of resonantly excited single-electron states. The formulae for the transition probabilities were derived in the LS coupling scheme, and the orbital angular momentum and spin selection rules were obtained. In addition to the formulae, which are exact in this order, three approximate expressions, which correspond to illustrative model mechanisms of the transition, were derived as limiting cases of the exact ones. Numerical results were obtained for the decay of the resonantly excited Kr 1 3d^{-1}5p[^1P] state which demonstrated quite clearly the important role of the interelectron interaction in double-ionization processes. On the other hand, the results obtained show that low-energy electrons can appear in the photoelectron spectrum below the ionization threshold of the 3d shell. As a function of the photon frequency, the yield of these low-energy electrons is strongly amplified by the resonant transition of the 3d electron to 5p (or to other discrete levels), acting as an intermediate state, when the photon frequency approaches that of the transition.

Nitrogen and carbon dynamics in grassland soils and plants after application of digestate

A better understanding of effects after digestate application on plant community, soil microbial community as well as nutrient and carbon dynamics is crucial for a sustainable grassland management and the prevention of species and functional diversity loss. The specific research objectives of the thesis were: (i) to investigate effects after digestate application on grass species and soil microbial community, especially focussing on nitrogen dynamic in the plant-soil system and to examine the suitability of the digestate from the “integrated generation of solid fuel and biogas from biomass” (IFBB) system as fertilizer (Chapter 3). (ii) to investigate the relationship between plant community and functionality of soil microbial community of extensively managed meadows, taking into account temporal variations during the vegetation period and abiotic soil conditions (Chapter 4). (iii) to investigate the suitability of IFBB-concept implementation as grassland conservation measure for meadows and possible associated effects of IFBB digestate application on plant and soil microbial community as well as soil microbial substrate utilization and catabolic evenness (Chapter 5). Taken together the results indicate that the digestate generated during the IFBB process stands out from digestates of conventional whole crop digestion on the basis of higher nitrogen use efficiency and that it is useful for increasing harvestable biomass and the nitrogen content of the biomass, especially of L. perenne, which is a common species of intensively used grasslands. Further, a medium application rate of IFBB digestate (50% of nitrogen removed with harvested biomass, corresponding to 30 50 kg N ha-1 a-1) may be a possibility for conservation management of different meadows without changing the functional above- and belowground characteristic of the grasslands, thereby offering an ecologically worthwhile alternative to mulching. Overall, the soil microbial biomass and catabolic performance under planted soil was marginally affected by digestate application but rather by soil properties and partly by grassland species and legume occurrence. The investigated extensively managed meadows revealed a high soil catabolic evenness, which was resilient to medium IFBB application rate after a three-year period of application.