On the analytic extension of three ratios of Horn’s con(cid:29)uent hypergeometric function H 7

. In this paper, we consider the extension of the analytic functions of two variables by special families of functions (cid:22) continued fractions. In particular, we establish new symmetric domains of the analytical continuation of three ratios of Horn’s con(cid:29)uent hypergeometric function H 7 with certain conditions on real and complex parameters using their continued fraction representations. We use Worpitzky’s theorem, the multiple parabola theorem, and a technique that extends the convergence, already known for a small domain, to a larger domain to obtain domains of convergence of continued fractions, and the PC method to prove that they are also domains of analytical continuation.


IF sntrodution
rypergeometri funtions elong to speil funtions tht ply n importnt role in solving wide lss of prolems in mthemtil physiE sD solid medium mehnisD eromehnisD quntum (eld theoryD proiE lity theoryD etF heir representtions in the form of hypergeometri seriesD whih usully hve limited regions of onvergeneD prompted the serh for their nlytil extension to wider regions using the integrl representtions or rnhed ontinued frtion expnsionsF xote tht hereD the region is domin @n open onneted suset of multidimensionl omplex speA together with llD prtD or none of its oundryF he prolem of onstruting n expnsion nd estlishing n nlytil extension ws onsidered in VDQI for generlized hypergeometri funtion p F q ; in IDIHDIRDPTDQH for eppel9s funtions F 1 !F 4 ; in PDWDIQDIVDPV for vuriell!rn9s funtions F D , F K , nd F S ; in QD IUD IW!PI for rorn9s hypergeometri funtions H 3 nd H 4 ; in RD U for rorn9s on)uent funtions H 6 nd H 7 ; in S for on)uent hypergeometri funtion Φ (N ) D .xote tht the estlished domins of nlytil ontinution in these works re the domins of onvergene of ontinued frtions or rnhed ontinued frtionsF his pper onsiders the rorn9s on)uent hypergeometri funtion H 7 de(E ned s follows @seeD PUA where α, γ 1 , γ 2 ∈ C, γ 1 , γ 2 ∈ {0, −1, −2, . ..},(a) k is the ohhmmer symol de(ned for ny omplex numer a nd nonEnegtive integer k y (a he pper R provides the following three expnsions where where sn dditionD it is shown tht is the domin of the nlytil ontinution of the funtion on the left side of @IFPA or @IFSA for eh α > −1 nd α = 0 or α > −1, α = 0, nd α = 1F end lso tht is the domin of the nlytil ontinution of the funtion on the left side of @IFRA for eh α > 0.
xote tht @IFTA is the grtesin produt of the plne ut long the rel xis from 1/16 to +∞ nd the plne ut long the rel xis from his pper ontinues the study of the domins of nlytil ontinution of the funtions on the left sides of @IFPAD @IFRAD nd @IFSAF wore out nlytil extension of the nlyti funtions y speil fmilies of funtions n e found in works P!RD TF PF gontinued frtions nd nlyti ontinution o prove our resultsD we inlude the following @see QPAF Theorem 1 @orpitzky9s heoremA.Let the elements d k , k ≥ 1, of the continued fraction lie in the circular region dened by Then: (A) The continued fraction (2.1) converges to a nite value.
(B) The values of the continued fraction (2.1) and of its approximants lie in the circular region |w − 1| ≤ 1/2.@PFPA end we lso need the following theorem PWD heorem RFRQF Theorem 2 @wultiple prol theoremA.Let the elements d k , k ≥ 1, of the continued fraction (2.1) lie in a parabolic regions dened by where p k > 0 and ψ k , k ≥ 0, are real, and where Then: (A) The continued fraction (2.1) converges to a nite value if there exists a constant The values of all approximants are nite and lie in the half plane xow we re redy to prove suh theoremF Theorem 3. Let α be a complex constant such that and where a 3k+2 and a 3k+3 , k ≥ 0, are dened in (1.3), α ∈ {0, −1, −2, . ..}, r is a positive number, {q k } is a sequence of real numbers such that where 0 < ε < 1/2.Then: (A) The continued fraction (1.2) converges uniformly on every compact subset of the domain where and (B) The function f (z) is an analytic continuation of the function on the left side of (1.2) in (2.6).
st follows from heorem P @fA tht for eh z ∈ Ω s,r , there exists hlf plne vet where 0 < δ < 1, e domin ontined in the domin Ω s,r .hen for the ritrry z ∈ Ω s,r,δ we otin for n ≥ 1 vet Ω e n ritrry ompt suset of Ω s,r .vet us over Ω with domins of form @PFIQAF prom this over we hoose the (nite suover hen for ritrry z ∈ Ω we otin |f n (z)| ≤ C(Ω), n ≥ 1, iFeF the sequene {f n (z)} is uniformly ounded on every ompt suset of Ω s,r .en pplitiE on of heorem Q Q yields the uniform onvergene of @IFPA to holomorphi funtions on ll ompt susets of Ω s,r .st is ler tht the ondition @PFWA holds for ll s > 0. vet now Ω e n ritrry ompt set ontined in @PFUAF hen Ω ⊆ Ω s,r ⊆ Ω r for some s su0iently smllD whose Ω s,r is n open iErdioid @PFIIAF his proves tht @eA holds in @PFUAF o prove @eAD it su0es to show tht this ssertion holds in @PFVAF st is ler tht the elements of @IFPA stisfy the onditions of heorem I @eAF his yields the onvergene of ontinued frtion @IFPA for ll z ∈ Ω α F st follows from heorem I @fA tht the pproximnts of @IFPA ll lie in the irulr region @PFPA if z ∈ Ω α F reneD y heorem Q QD the onvergene of the ontinued frtion @IFPA is uniform on every ompt suset of the domin @PFVAF he proof of @fA is nlogous to the proof of heorem Q @fA TY hene it is omittedF he grphil illustrtions of domins for vriles z 1 nd z 2 from @PFTA is given in pigure I @A!@AF Re(z 1 ) Im(z 1 )  he following result gives the nlytil extension domin tht is the grtesin produt of the plne ut long the rel xis from −1/16 to −∞ nd the plne ut long the rel xis from −(1 + α)/4] to −∞, where arg(z 1 ) = arg(z 2 ) nd α is rel onstnt tht stis(es the following onditions @PFIRAF Theorem 4. Let α be a real constant such that where a 3k+2 and a 3k+3 , k ≥ 0, are dened in (1.3), a is a positive number.
Then the continued fraction (1.2) converges uniformly on every compact subset of the domain to a function f (z) holomorphic in Ω r,α , and f (z) is an analytic continuation of the function on the left side of (1.2) in (2.15).