Polyurethane reactive hot-melt adhesives (PURHs) are frequently employed in industries; however, there is still a need to develop more sustainable and versatile methodologies to expand the functions and fabrication of these important materials. Renewable feedstock can give PURHs with new functions, and reduce environmental impact. This study focuses on synthesizing PURHs using polyols derived from biomass (plants) and greenhouse gas (CO 2 ) resources. These PURHs were characterized by multiple techniques, including solid-state 13 C nuclear magnetic resonance (NMR), a dynamic mechanical analysis (DMA), single-lap adhesive joints strength of stainless steel, and hydrolytic ageing. The PURH film based on biomass poly(tetramethylene ether) glycol (bio-PTMEG) exhibited better water vapor permeability, tensile strength, and adhesive joints properties than PURHs based on cashew nutshell liquid (CNSL) polyester diol and poly(propylene carbonate)-poly(propylene glycol) (PPC-PPG) copolymer diol. The polyols blend of bio-PTMEG with biomass and CO 2 based polycarbonate diols respectively provided PURHs films excellent hydrolysis resistance and adhesive strength on single-lap adhesively bonded stainless steel specimens. The work herein demonstrates that various renewable polyols can be employed in a sustainable fashion to optimize the structures and properties of PURHs for important applications.Abstract: Let F denote a eld and let V denote a vector space over F with nite positive a pair A, A * of diagonalizable F-linear maps on V, each of which acts on an eigenbasis irreducible tridiagonal fashion. Such a pair is called a Leonard pair. We consider the s there exists an automorphism of the endomorphism algebra of V that swaps A and A * . S is unique, and called the duality A ↔ A * . In the present paper we give a comprehens duality. In particular, we display an invertible F-linear map T on V such that the map X → A ↔ A * . We express T as a polynomial in A and A * . We describe how T acts on ag and 24 bases for V.