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In this paper, we study a generalization of Narayana’s numbers and Padovan’s numbers. This generalization also includes a sequence whose elements are Fibonacci numbers repeated three times. We give combinatorial interpretations and a graph interpretation of these numbers. In addition, we examine matrix generators and determine connections with Pascal’s triangle.

Somos conjectured thousands of Dedekind $$\eta $$ -function identities of various levels, around 6200 in number. He did so using computational evidence but has not sought to provide any proof for these identities. In this paper, we prove level 6 of Somos’s Dedekind $$\eta $$ -function identities containing five terms in two methods. Further, as an application of these identities, we deduce colored partitions for the same.

In this paper, we first give a new proof of the log-Minkowski inequality of general planar convex bodies and then extend the $$L_p$$ -Brunn–Minkowski inequality and $$L_p$$ -Minkowski inequality of o-symmetric planar convex bodies for $$p\in (0,1)$$ to $$\phi $$ -Brunn–Minkowski inequality and $$\phi $$ -Minkowski inequality of general planar convex bodies. As an application, a family of $$\phi $$ -measures of asymmetry for planar convex bodies is introduced.

Let $$\{V_n(P,Q)\}$$ be the Lucas sequence of the second kind at the nonzero relatively prime parameters P and Q. In this paper, we present techniques for studying the solutions (x, n) with $$x \ge 2, n \ge 0$$ of any Diophantine equation of the form $$\begin{aligned} \frac{1}{V_n(P_2,Q_2)}=\sum _{k=1}^{\infty }\frac{V_{k-1}(P_1,Q_1)}{x^k} \end{aligned}$$ in both cases $$(P_1,Q_1)=(P_2,Q_2)$$ and $$(P_1,Q_1)\ne (P_2,Q_2)$$ , where $$Q_1, Q_2 \in \{-1,1\}$$ . Furthermore, we represent the procedures of these techniques in case of $$ -2 \le P_1, P_2 \le 4$$ .

In this paper, we study a schistosomiasis model incorporating the miracidia and cercariae dynamics, discrete-time delays as well as control measures like water treatment. Modelling the dynamics of schistosomiasis infectious disease is quite challenging because of the different larval forms assumed by the parasite and the requirement of two hosts during the life cycle. Our model is generic in the sense that it considers both situations where particle depletion by hosts or snails could have or not a negligible impact on particle dynamics. Precisely, we introduce two parameters u and v such that when $$u=v=0$$ , then particle depletion by hosts or snails is not considered; when $$u=v=1$$ , then particle depletion by hosts and snails is considered. The model is analyzed to gain insights into the qualitative features of the disease-free equilibrium which allows the determination of the basic reproductive number $${\mathcal {R}}_{u,v}$$ . The Center Manifold Theory is used to discuss existence and local stability of an endemic equilibrium. Global sensitivity analysis (SA) of the schistosomiasis model and the basic reproduction number are carried out. SA results of the model point out the leading role of $$\eta $$ , the parameter that shapes infection-induced death rate in humans, on the dynamics of humans (susceptible and infected), miracidia, cercariae and infected snails. They also reveal the pervasive role of $$\theta $$ , the water treatment-induced death rate of snails, on the dynamics of infected humans, miracidia, snails (susceptible and infected) and cercariae. SA results of the basic reproduction number highlight the role of $$\eta $$ , $$\theta $$ , $$\lambda $$ (the contact rate of transmission of miracidia to susceptible snails), $$\varpi $$ (production rate of miracidia from feces of infected humans) and $$\gamma $$ (the transmission rate of cercariae to susceptible humans). Therefore, a possible way to control the disease could rely on the intensification of sanitization campaigns that will result in an increase of $$\theta $$ together with sensitization about the necessity to have a treatment once you are infected to reduce $$\eta $$ .

In this paper, we determine all repdigits in base b for $$2\le b\le 10,$$ which are products of two Pell numbers or Pell–Lucas numbers. It is shown that the largest Pell number which is a base b-repdigit is $$P_{6}=70=(77)_{9} =7+7\cdot 9.$$ Also, we give the result that the equations $$P_{m}P_{n}+1=b^{k}$$ and $$Q_{m}Q_{n}+1=b^{k}$$ have no solutions for $$n\ge 5$$ and $$n\ge 1,$$ respectively, where $$1\le m\le n$$ .