Explicit formulas for Euler polynomials and Bernoulli numbers

In this paper, we give several explicit formulas involving the n-th Euler polynomial E_{n}\left(x\right). For any fixed integer m\geq n, the obtained formulas follow by proving that E_{n}\left(x\right) can be written as a linear combination of the polynomials x^{n}, \left(x+r\right)^{n},\ldots, \left(x+rm\right)^{n}, with r\in \left \{1,-1,\frac{1}{2}\right\}. As consequence, some explicit formulas for Bernoulli numbers may be deduced.

New Generalized Apostol-Frobenius-Euler polynomials and their Matrix Approach

In this paper, we introduce a new extension of the generalized Apostol-Frobenius-Euler polynomials ℋn[m−1,α](x; c,a; λ; u). We give some algebraic and differential properties, as well as, relationships between this polynomials class with other polynomials and numbers. We also, introduce the generalized Apostol-Frobenius-Euler polynomials matrix ????[m−1,α](x; c,a; λ; u) and the new generalized Apostol-Frobenius-Euler matrix ????[m−1,α](c,a; λ; u), we deduce a product formula for ????[m−1,α](x; c,a; λ; u) and provide some factorizations of the Apostol-Frobenius-Euler polynomial matrix ????[m−1,α](x; c,a; λ; u), which involving the generalized Pascal matrix.

Research and application of novel Euler polynomial-driven grey model for short-term PM10 forecasting

PurposePM10 is one of the most dangerous air pollutants which is harmful to the ecological system and human health. Accurate forecasting of PM10 concentration makes it easier for the government to make efficient decisions and policies. However, the PM10 concentration, particularly, the emerging short-term concentration has high uncertainties as it is often impacted by many factors and also time varying. Above all, a new methodology which can overcome such difficulties is needed.Design/methodology/approachThe grey system theory is used to build the short-term PM10 forecasting model. The Euler polynomial is used as a driving term of the proposed grey model, and then the convolutional solution is applied to make the new model computationally feasible. The grey wolf optimizer is used to select the optimal nonlinear parameters of the proposed model.FindingsThe introduction of the Euler polynomial makes the new model more flexible and more general as it can yield several other conventional grey models under certain conditions. The new model presents significantly higher performance, is more accurate and also more stable, than the six existing grey models in three real-world cases and the case of short-term PM10 forecasting in Tianjin China.Practical implicationsWith high performance in the real-world case in Tianjin China, the proposed model appears to have high potential to accurately forecast the PM10 concentration in big cities of China. Therefore, it can be considered as a decision-making support tool in the near future.Originality/valueThis is the first work introducing the Euler polynomial to the grey system models, and a more general formulation of existing grey models is also obtained. The modelling pattern used in this paper can be used as an example for building other similar nonlinear grey models. The practical example of short-term PM10 forecasting in Tianjin China is also presented for the first time.

Truncated euler polynomials

We define a truncated Euler polynomial Em,n(x) as a generalization of the classical Euler polynomial En(x). In this paper we give its some properties and relations with the hypergeometric Bernoulli polynomial.

WASD Algorithm with Pruning-While-Growing and Twice-Pruning Techniques for Multi-Input Euler Polynomial Neural Network

Differing from the conventional back-propagation (BP) neural networks, a novel multi-input Euler polynomial neural network, in short, MIEPNN (specifically, 4-input Euler polynomial neural network, 4IEPNN) is established and investigated in this paper. In order to achieve satisfactory performance of the established MIEPNN, a weights and structure determination (WASD) algorithm with pruning-while-growing (PWG) and twice-pruning (TP) techniques is built up for the established MIEPNN. By employing the weights direct determination (WDD) method, the WASD algorithm not only determines the optimal connecting weights between hidden layer and output layer directly, but also obtains the optimal number of hidden-layer neurons. Specifically, a sub-optimal structure is obtained via the PWG technique, then the redundant hidden-layer neurons are further pruned via the TP technique. Consequently, the optimal structure of the MIEPNN is obtained. To provide a reasonable choice in practice, several different MATLAB computing routines related to the WDD method are studied. Comparative numerical-experiment results of the 4IEPNN using these different MATLAB computing routines and the standard multi-layer perceptron (MLP) neural network further verify the superior performance and efficacy of the proposed MIEPNN equipped with the WASD algorithm including PWG and TP techniques in terms of training, testing and predicting.