Investigation of Time Scaling for the Inverted Beta Functions

Abstract

The use of Ateb-functi­ons is de­ter­mi­ned by tho­se are­as whe­re or­di­nary tri­go­no­met­ric functi­ons are used. Mo­dern ad­van­ces in physics ha­ve led to the de­ve­lop­ment of new mat­he­ma­ti­cal are­as that req­ui­re the re­la­ti­vity or va­ri­abi­lity of ti­me. The cur­rent re­se­arches in this fi­eld and ma­in re­sults of stu­di­es of the or­di­nary Ateb functi­ons are bri­efly descri­bed. To ta­ke in­to ac­co­unt compres­si­on/slow-down as a pro­perty of ti­me pa­ra­me­ter, the q-ana­logs of Ateb-si­ne (q-Ateb-si­ne) and Ateb-co­si­ne (q-Ateb-co­si­ne) are construc­ted by in­ver­ting the in­comple­te q-Be­ta functi­ons. The chan­ge in pa­ra­me­ter q cor­res­ponds to the ti­me sca­ling in the stu­di­es. q-ana­logs of Ateb-tan­gent (q-Ateb-tan­gent), Ateb-co­tan­gent (q-Ateb-co­tan­gent), Ateb-se­cant (q-Ateb-se­cant) and Ateb-co­se­cant (q-Ateb-co­se­cant) are intro­du­ced. The­orems cha­rac­te­ri­zing the ba­sic pro­per­ti­es of the construc­ted functi­ons are pro­ved. In par­ti­cu­lar, it is shown that when q→1, ta­king the li­mit we ob­ta­in or­di­nary Ateb-functi­ons. The intro­du­ced functi­ons are pe­ri­odic with the pe­ri­od cor­res­pon­ding to q-ana­log­ue pe­ri­ods of the or­di­nary Ateb-functi­ons. The rep­re­sen­ta­ti­on of the pe­ri­od using the q-ana­log­ue of the Gam­ma-functi­on is construc­ted. The ge­ne­ra­li­zed Pytha­go­re­an iden­tity for the q-ana­log­ues of tri­go­no­met­ric Ateb-functi­ons is pro­ved. Al­so the pro­per­ti­es of the pa­rity and od­dity of the­se functi­ons are con­si­de­red and pro­ved. The in­ter­vals of incre­asing/dec­re­asing for all functi­ons are fo­und. The q-ana­log­ues of the iden­ti­ti­es for­mu­las for the tri­go­no­met­ric Ateb-functi­ons are pre­sen­ted. For­mu­las for cal­cu­la­ting q-de­ri­va­ti­ves for the q-ana­log­ue of tri­go­no­met­ric Ateb-functi­ons are construc­ted. It is pro­ved that construc­ted functi­ons sa­tisfy the system of q-de­ri­va­ti­ve dif­fe­ren­ti­al eq­ua­ti­ons. Re­sults of the pre­sen­ted stu­di­es can be used in the ti­me se­ri­es the­ory and sig­nal pro­ces­sing.