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Yi, Jinhee

doi:10.1023/a:1013991704758

Cited by 10 publications

(6 citation statements)

References 10 publications

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“…4 . After injecting in this 'large' non-linear differential equation, equation (11), the Schwarzian condition (28) with W(x) given by (29), and the Calabi-Yau condition (31), we eventually find that this last 'large' equation becomes independent of the pullback y(x) and reduces to a quite simple condition giving s(x) as a polynomial expression in the two coefficients p(x) and q(x) and their derivatives:…”

Section: Calabi-yau Condition (Exterior Square)mentioning

confidence: 98%

“…Our large calculations thus show that the pullback-compatibility of an order-four linear differential operator which verifies the Calabi-Yau condition (31), amounts to saying that this order-four linear differential operator reduces to (the symmetric cube of) an underlying ordertwo linear differential operator. The Schwarzian condition (28) with W(x) given by (29), is thus inherited from the Schwarzian condition (9) of the underlying ordertwo linear differential operator.…”

Section: Calabi-yau Condition (Exterior Square)mentioning

confidence: 99%

“…4 for an order-four linear differential operator, corresponding to this parametrization (such that it verifies the symmetric Calabi-Yau condition, and such that its symmetric square is of order nine), one naturally finds the Schwarzian condition ( 28) with ( 29), as well as the exact expression (30) for the conjugation function v(x). Taking into account the Schwarzian condition (28), the identification of the coefficients of D x for L ( p)…”

Section: Symmetric Calabi-yau Conditionmentioning

confidence: 99%

“…In general, finding the Schwarzian equation ( 87) is easy, and getting solutions order by order as series expansions is also easy. However finding the selected values of the rational numbers a n such that the differentially algebraic [3,4] series y n (a n , x) are globally bounded and thus can be algebraic functions, and, possibly, modular correspondences, is a quite difficult task 28 .…”

Section: A Pre-modular Conceptmentioning

confidence: 99%

“…For modular correspondences see also the concept of modular equations[25][26][27][28] 6. The D 3x coefficient is normalized to 1.…”

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confidence: 99%

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Schwarzian conditions for linear differential operators with selected differential Galois groups

Abdelaziz¹,

Maillard²

2017

J. Phys. A: Math. Theor.

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Abstract. We show that non-linear Schwarzian differential equations emerging from covariance symmetry conditions imposed on linear differential operators with hypergeometric function solutions can be generalized to arbitrary order linear differential operators with polynomial coefficients having selected differential Galois groups. For order three and order four linear differential operators we show that this pullback invariance up to conjugation eventually reduces to symmetric powers of an underlying order-two operator. We give, precisely, the conditions to have modular correspondences solutions for such Schwarzian differential equations, which was an open question in a previous paper. We analyze in detail a pullbacked hypergeometric example generalizing modular forms, that ushers a pullback invariance up to operator homomorphisms. We expect this new concept to be well-suited in physics and enumerative combinatorics. We finally consider the more general problem of the equivalence of two different orderfour linear differential Calabi-Yau operators up to pullbacks and conjugation, and clarify the cases where they have the same Yukawa couplings.

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Section: Calabi-yau Condition (Exterior Square)mentioning

confidence: 98%

Section: Calabi-yau Condition (Exterior Square)mentioning

confidence: 99%

Section: Symmetric Calabi-yau Conditionmentioning

confidence: 99%

Section: A Pre-modular Conceptmentioning

confidence: 99%

“…For modular correspondences see also the concept of modular equations[25][26][27][28] 6. The D 3x coefficient is normalized to 1.…”

mentioning

confidence: 99%

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Schwarzian conditions for linear differential operators with selected differential Galois groups

Abdelaziz¹,

Maillard²

2017

J. Phys. A: Math. Theor.

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The Classical Special Functions

The Mathematica GuideBook for Symbolics

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A new proof of Ramanujan’s modular equation relating R(q) with R(q 5)

Gugg

2009

Ramanujan J

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We give a new proof of Ramanujan's modular identity relating R(q) with R(q 5 ), where R(q) is the famous Rogers-Ramanujan continued fraction. Our formulation is stronger than those of preceding authors; in particular, we give for the first time identities for the expressions appearing in the numerator and the denominator of Ramanujan's identity. A related identity for R(q) that has partition-theoretic connections is also proved.

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Untitled

Cited by 10 publications

References 10 publications

Schwarzian conditions for linear differential operators with selected differential Galois groups

Schwarzian conditions for linear differential operators with selected differential Galois groups

The Classical Special Functions

A new proof of Ramanujan’s modular equation relating R(q) with R(q 5)

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