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_ jacobi - type continued fractions _ , or _ j - fractions _ , correspond to power series defined by infinite continued fraction expansions of the form @xmath7 for some sequences @xmath8 and @xmath9 , and some ( typically formal ) series variable @xmath10 @xcite @xcite . the formal series enumerated by special cases of the truncated and infinite j - fraction series of this form include _ ordinary _ ( as opposed to typically closed - form _ exponential _ ) _ generating functions _ for many one and two - index combinatorial sequences including the generalized factorial functions studied in the references and in the results from flajolet s articles . the references provide infinite formal continued fraction series for the typically divergent ordinary generating functions of many special sequences , factorial functions , and combinatorial triangles arising in applications @xcite .    [ [ j - fraction - expansions - of - ordinary - square - series - generating - functions ] ] j - fraction expansions of ordinary square series generating functions ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~    in this article , we consider a @xmath1-series - related variant of the j - fractions in enumerating the square - power terms of @xmath0 , which we have discovered computationally ( see @xcite for integral representations ) . in particular , for non - zero @xmath11 such that @xmath12 , we define the following jacobi - type j - fraction , @xmath13}(q , z)$ ] , for the ordinary square series generating function defined in @xcite as follows : @xmath14}(q , z ) & =        \cfrac{1}{1-qz-\cfrac{q^2(q^2 - 1 ) z^2}{1-q(q^4+q^2 - 1)z-        \cfrac{q^8 ( q^4 - 1 ) z^2}{1-q^3(q^6+q^4 - 1)z-\cfrac{q^{14}(q^6 - 1 ) z^2}{\cdots . } } } } \end{aligned}\ ] ]    [ def_ciq_abiq_phqz_qhqz ] we define the component sequence functions , @xmath15}(q)$ ] and @xmath16}(q)$ ] , as follows for @xmath17 : @xmath18}(q ) & : =        \begin{cases }             q^{2h-3 } \cdot \left(q^{2h } + q^{2h-2 } - 1\right ) &             \text { if $ h \geq 2 $ ; } \\             q & \text { if $ h = 1 $ ; } \\             1 & \text { if $ h = 0 $ , }        \end{cases } \\   \label{eqn_abhq_subseq_def }   \operatorname{ab}_h^{[\sq]}(q ) & : =        \begin{cases }             q^{6h-10 } \cdot \left(q^{2h-2 } - 1\right ) &             \text { if $ h \geq 2 $ ; } \\             0 & \text { otherwise . }       \end{cases } \end{aligned}\ ] ] we then define the @xmath3 convergent functions , @xmath19 , recursively through the component numerator and denominator functions given by , as @xmath20 . similarly , @xmath21 in the remainder of the article . ] @xmath22}(q ) \cdot z ) \operatorname{p}_{h-1}(q , z ) -        \operatorname{ab}_{h}^{[\sq]}(q ) \cdot z^2 \operatorname{p}_{h-2}(q , z ) + \iverson{h = 1 } \\ \notag   \operatorname{q}_h(q , z ) & = ( 1-c_{h}^{[\sq]}(q ) \cdot z ) \operatorname{q}_{h-1}(q , z ) - \operatorname{ab}_{h}^{[\sq]}(q ) \cdot z^2 \operatorname{q}_{h-2}(q , z ) +        ( 1-c_{1}^{[\sq]}(q ) \cdot z ) \iverson{h = 1 } + \iverson{h = 0}. \end{aligned}\ ] ]    we claim , and subsequently prove in section [ section_proofs ] , that the truncated @xmath23-order power series expansions of the @xmath3 convergent functions , @xmath24 , defined in definition [ def_ciq_abiq_phqz_qhqz ] generate the so - termed `` _ _ square series _ _ '' powers defined in @xcite as @xmath25 for all @xmath26 . more precisely , we state the next result as our main theorem which we carefully prove in section [ subsection_proofofmaintheorem ] of the article below .      for @xmath30 , we show that the limiting convergent function , denoted by @xmath31}(q , z )        : = \sum_{n \geq 0 } q^{n^2 } z^n =        \lim_{h \longrightarrow \infty}\ \conv_h(q , z ) , \end{aligned}\ ] ] which corresponds to the integral representations of the _ ordinary geometric square series _ cases defined in @xcite given by @xmath32}(q , z ) & = \int_0^{\infty }        \frac{2 e^{-t^2/2}}{\sqrt{2\pi } } \left [        \frac{1 - z \cosh\left(t \sqrt{2 \log(q)}\right ) } {        z^2 - 2z \cosh\left(t \sqrt{2 \log(q)}\right ) + 1 }        \right ] dt , \end{aligned}\ ] ] exists , and that the limiting cases of this convergent function series converge uniformly as a function of @xmath33 with respect to the sequence of @xmath3 partial sums of defined below . thus we are able to prove new infinite @xmath1-series identities generating special cases of @xmath13}(q , c \cdot q^{p})$ ] in section [ section_proofs ] , which arise in many applications to theta functions and special function series . the primary new form of the infinite q - series expansions we prove in section [ section_proofs ] of the article has the form @xmath34}(q , z ) & = \sum_{i=1}^{\infty }        \frac{(-1)^{i-1 } q^{(3i-4)(i-1 ) } { \left(q^2 ; q^2\right)_{i-1 } } z^{2i-2 } } {        \sum\limits_{0 \leq j \leq n < 2i }        { \gkpsi{i}{j}_{q^2 } } { \gkpsi{i-1}{n - j}_{q^2 } } q^{2j } \cdot        \left(-q^{2i-3 } \cdot z\right)^n}. \end{aligned}\ ] ] moreover , the @xmath3 partial sums of the infinite @xmath1-series expansion in corresponding to the @xmath3 convergent function , @xmath24 , is always at least @xmath23-order accurate in generating the coefficients , @xmath0 , of @xmath35 for @xmath28 . in addition , we show that we can differentiate these new infinite square series expansions termwise with respect to @xmath33 thus allowing us to generate series for higher - order derivatives of the _ jacobi theta functions _ , which are generated as particular cases of the geometric square series , @xmath36 , in @xcite and which similarly arise in many well - known identities and classical expansions of special products and partition function generating functions . more precisely , if we let @xmath38 denote an arbitrary sequence depending on the fixed parameter @xmath1 , we seek to explore the analogous @xmath1-series - related j - fraction expansions enumerating this sequence in the form of @xmath39 we can always solve for the sequences of implicit coefficients , @xmath40 and @xmath41 , in which ( at least formally ) generate the sequence of @xmath42 over @xmath33 . typically we find that for more well - known @xmath1-series generating function forms it is easy to computationally `` _ _ guess _ _ '' exact formulas for these sequences , which are each implicit functions of the @xmath43 s , in many special case forms . we conjecture , but do not offer conclusive proofs , that we have related @xmath1-series generating functions enumerating variants of the sequences , @xmath44 , @xmath45 , and @xmath46 over @xmath33 defined by the implicit subsequence listings given in table [ table_conclusions_specialj - fraction_conjsubseq_results ] . examples of special series that result by generating the sequences , @xmath42 , in the table by the properties resulting from our conjectured j - fraction expansions include the following expansions @xcite @xcite : @xmath47 \\   \tag{$q$-exponential function }   e_q(z ) & = \sum_{n=0}^{\infty } \frac{(1-q)^n z^n}{{\left(q ; q\right)_{n } } } \\   \tag{$q$-hypergeometric function }   _ 1\phi_0(0 ; - ; q , z ) & = \sum_{n=0}^{\infty }        \frac{z^n}{{\left(q ; q\right)_{n } } } =        \frac{1}{{\left(z ; q\right)_{\infty } } } \\   \tag{partition function generating function }   _ 1\phi_0(0 ; - ; q , q ) & = \sum_{n=0}^{\infty }        \frac{q^n}{{\left(q ; q\right)_{n } } } =        \frac{1}{{\left(q ; q\right)_{\infty } } } \\   \tag{another $ q$-binomial theorem }   _ 1\phi_0(0 ; - ; q , q ) & = { \left(z q^{-n } ; q\right)_{n}}. \end{aligned}\ ] ] for the first sequence case in the table , we can extend the methods of proof given in this article for the square series generating functions to find that @xmath48 other examples of new infinite series and q - series expansions for the generating functions of the sequences listed in table [ table_conclusions_specialj - fraction_conjsubseq_results ] , along with many other special cases not considered within the context of the applications cited in this article , are proved similarly .
we consider new series expansions for variants of the so - termed ordinary geometric square series generating functions originally defined in the recent article titled `` _ _ square series generating function transformations _ _ '' ( arxiv : 1609.02803 ) . whereas the original square series transformations article adapts known generating function transformations to construct integral representations for these square series functions enumerating the square powers of @xmath0 for some fixed non - zero @xmath1 with @xmath2 , we study the expansions of these special series through power series generated by jacobi - type continued fractions , or j - fractions 
 . we prove new exact expansions of the @xmath3 convergents to these continued fraction series and show that the limiting case of these convergent generating functions exists as @xmath4 . 
 we also prove new infinite @xmath1-series representations of special square series expansions involving square - power terms of the series parameter @xmath1 , the @xmath1-pochhammer symbol , and double sums over the @xmath1-binomial coefficients . 
 applications of the new results we prove within the article include new @xmath1-series representations for the ordinary generating functions of the special sequences , @xmath5 , and @xmath6 , as well as parallels to the examples of the new integral representations for theta functions , series expansions of infinite products and partition function generating functions , and related unilateral special function series cited in the first square series transformations article .    : _ square series ; @xmath1-series ; j - fraction ; continued fraction ; sum of squares functions ; sum of divisors function ; theta function ; ordinary generating function . _    : _ 05a15 ; 11y65 ; 11b65 ; 40a15 . _