The term 'periphrasis' broadly refers to constructions in which a syntactic sequence of two (or more, at least in principle) words can be considered to represent the instantiation of a particular morphological form of a single lexeme. For example, in English the sequences is playing and has played can be considered to instantiate the present progressive and present perfect tenses, respectively, of the verb play, alongside the single-word simple present and past forms, respectively plays and played.
Much recent work on periphrasis has adopted a fundamentally morphological approach to the phenomenon, often within a paradigmatic approach to the lexicon and morphological systems. Within such an approach, whenever a particular slot or set of slots in a morphological paradigm are filled by multiword sequences, we are dealing with periphrasis. An important question is then how to model this, given that we are dealing with single paradigm slots filled by (usually) two words, one of which words is usually identical with a word from a different morphological paradigm. For example, English plays unproblematically fills the paradigm slot of 3sg simple present for the verb play, and on some level or other this form can be treated as comprising the base form of the verb followed by a morpheme -s, which in English characterizes this particular paradigm slot. The 3sg present progressive is playing, on the other hand, involves not only a morphologically specific form of the lexical verb, play-ing, but also a form of a different verb, the 3sg simple present of be.
Different ways of modelling this are conceivable. If one simply assumes that English present progressive paradigms come fully specified as two-word sequences, am playing, are playing, is playing, etc., then the clear identity relation between the first word of each sequence and the corresponding forms of the simple present of be cannot be captured: is playing fills the slot for 3sg present progressive of play, and is fills the slot for 3sg simple present of be, but there is no representation of the fact that the latter is a part of the former. Alternatively, then, one could assume that the slots of the English progressive paradigm are filled simply by the form playing, parallel therefore to the single words filling non-periphrastic slots of the wider verbal paradigm, but that these forms come with a strict requirement that a corresponding form from another paradigm (in this case the simple present of be) must co-occur with them.
For further discussion of and references to work on periphrasis see Spencer (2006) and Lowe (2017).
2. Periphrasis in Sanskrit
The Sanskrit verbal system includes two formations which are traditionally (in the West) labelled 'periphrastic': the periphrastic future and the periphrastic perfect.
The status of the periphrastic future is complex; it is argued not to be a true periphrasis by Lowe (2017). Historically, at least, the periphrastic future evidently derives from a collocation of an agent noun in -tṛ with an auxiliary form of the copula as 'be'. So, to the root kṛ 'do, make', the 1sg active periphrastic future, kartāsmi, is apparently simply derived from the sandhi of the nominative singular of the agent noun kartṛ- 'doer', i.e. kartā, with the 1sg. present of as, i.e. asmi: kartā + asmi → kartāsmi. However, other forms of the paradigm are less transparent. For example, first and second person dual and plural forms appear to show the singular of the agent noun, rather than dual or plural. So, 1pl. kartāsmaḥ, apparently from nom.sg. kartā + 1pl. smaḥ, rather than from nom.pl. kartāraḥ + smaḥ as we might expect. In some forms of Sanskrit, however, in particular the language of the Sanskrit epics, unambiguous two-word sequences like kartāraḥ smaḥ, with expected number agreement, are found.
Lowe (2017) shows that there are two 'dialects' of Sanskrit in regard to the treatment of the periphrastic future. In one, the periphrastic future is not necessarily anything more than a syntactic collocation of agent noun plus form of as 'be', and synthetic forms like kartāsmaḥ are never found. In contrast, in the other the periphrastic future is entirely non-periphrastic: it is a purely synthetic paradigm, with unambiguously two-word sequences like kartāraḥ smaḥ unattested. The Indian grammatical tradition, following Pāṇini, treats the periphrastic future in the latter manner, that is as an entirely synthetic, non-periphrastic, paradigm, and there is therefore nothing in its analysis which can speak to how the Indian grammatical tradition treated periphrasis.
The periphrastic perfect is, by comparison, a remarkably clear case of inflectional periphrasis. The inherited synthetic perfect tense, due to the specific morphology of its formation, cannot be used with derived verbal stems (causatives, desideratives, intensives and denominatives) nor, for historical reasons, with a number of basic verbal roots. The periphrastic perfect fills this gap for these verbs / verbal stems; it is comprised of an invariant form of the lexical stem, ending in -ām (likely in origin accusative of an abstract noun), and an auxiliary, either the perfect of the root kṛ 'do, make' (exclusively in the earlier language), or the root as 'be' (most commonly in the later language), or the root bhū 'become' (never in the earlier language, and rare later). For example, to the causative of the verb gam 'go', i.e. gamaya-ti 'makes go', the 3sg. perfect is gamayāṃ cakāra 'made go', where cakāra is 3sg perfect of kṛ 'do, make' (or alternatively gamayām āsa or gamayāṃ babhūva, mutatis mutandis). The periphrastic perfect shows a number of features which prove it cannot be treated as either a purely syntactic sequence (such as the fact that forms like gamayām have no independent existence), nor as a purely morphological sequence (since in early texts the two elements are sometimes found separated). It is therefore an ideal example of a true periphrasis. The periphrastic perfect is discussed further by Ozono (2016).
3. Periphrasis in the Aṣṭādhyāyī
As mentioned above, Pāṇini's Aṣṭādhyāyī shows no recognition that the so-called 'periphrastic future' has any periphrastic properties. In Pāṇini's system, the 'periphrastic future' is treated as a separate inflectional tense/mood formation with abstract marker luṭ (contrasting with present laṭ, aorist luṅ, etc.). The marker luṭ conditions the introduction of a suffix -tās (Aṣṭ 3.1.33), entirely parallel to the tense/mood markers of the unambiguously synthetic tenses and moods. The peculiarities of the inflectional endings of the periphrastic future, which reflect its origin in a collocation of agent noun plus copula, are treated by means of simple substitution rules (Aṣṭ 2.4.85, 7.4.50—52), in the same way that, for example, the endings specific to the perfect are derived by substitution of the 'basic' endings, and without any reference whatsoever to either the form of the agent noun in -tṛ or to any forms of the verb as 'be'. The only way in which the periphrastic future differs from any of the other synthetic tenses and moods, for Pāṇini, is that it does not behave like other finite verbs in losing its accent when non-initial in main clauses (Aṣṭ. 8.1.29). This historically reflects its origin in an expression containing a noun (which would not lose its accent in this context), but for Pāṇini is nothing more than an exception to the general rule for synthetic finite verbs.
With the periphrastic perfect, the situation is different. Aṣṭ. 3.1.35–39 state the introduction of the suffix -ām after the relevant set of verbs / verbal stems when followed by the perfect marker liṭ. So for example:
Crucially, Aṣṭ 3.1.40 introduces the verb kṛ 'do, make' (Pāṇini's kṛÑ) following this ām. The rule uses the word anuprayujyate 'is employed in addition after', which is here used to mark this use of kṛ as distinct from other ordinary uses of the verb. At Aṣṭ. 1.3.63 this 'additional employment' of kṛ is referred to using the associated noun, anuprayoga, ensuring that the scope of that rule applies only to kṛ used (in our terms) as an auxiliary in the periphrastic perfect, and not to kṛ in other contexts. This is the closest Pāṇini comes to an explicit recognition of periphrastic expression.
This introduction of kṛ is specified as occurring before liṭ, which implies the introduction of the sequence kṛ + liṭ, which will produce the perfect forms of this verb. The liṭ which originally follows the lexical stem is deleted by Aṣṭ. 2.4.81. (The introduction of a new, and deletion of the old, liṭ is necessary to ensure the correct scope of further morphological operations, such as reduplication, conditioned by liṭ.) We therefore have the following derivation:
Following the ordinary rules of suffixation, substitution and sandhi, this will produce a periphrastic form such as gamayāṃ cakāra. The processes which apply to derive the form of the auxiliary are exactly the same processes which apply to kṛ when used as a lexical verb. Aṣṭ. 1.3.63, mentioned above, is the only rule which distinguishes this auxiliary use of kṛ. It is an important rule for the periphrastic status of the construction, in that it specifies that the voice of kṛ when used in this construction depends on the voice requirements of the lexical verb. For example, the root ās 'sit' is a deponent verb, occurring only in the mediopassive voice. In the periphrastic perfect, this voice requirement is realised on the auxiliary: āsāṃ cakre, with 3sg. mediopassive perfect cakre, but never with active 3sg. * āsāṃ cakāra.
In comparison with modern approaches to periphrasis, Pāṇini's treatment of the periphrastic perfect is similar to the approach sketched above, whereby the 'paradigmatic' form of the periphrastic perfect itself consists only of the form of the lexical stem itself, e.g. gamayām, but this comes with a collocation specification requiring the presence of the perfect of kṛ.
It is not, however, clear whether Pāṇini intended to license any positional freedom for the two elements of the periphrastic perfect. The wording of Aṣṭ. 3.1.40 implies that the form of kṛ follows the suffix ām, and this is standardly interpreted as implying that it directly follows, i.e. that it is not possible for other words to intervene between the two forms. This understanding of the sense of anuprayujyate is first found in Kātyāyana. It is at least possible, however, that just as with the addition of as and bhū as auxiliaries this reflects not the original intention of Pāṇini, but the later grammar of the construction in Kātyāyana's form of the language. The matter requires further investigation.
 The non-Pāṇinian Kātantra (K. 3.1.30) goes even further, treating the suffix+ending complexes of the periphrastic future as unsegmentable units, e.g. 2sg. -tāsi in place of -tās + -si etc.
 Following Kātyāyana and Patañjali, the tradition takes Pāṇini's reference to the root kṛ in this case to include reference also to the roots as 'be' and bhū 'become'. The fact that they felt this was necessary shows that as and bhū were valid auxiliaries in the periphrastic perfect in their form of the language, but this does not necessarily mean that the same was the case for Pāṇini. The most likely explanation for Pāṇini's referring only to kṛ is that for him, as in the pre-Classical language more generally, this was the only possible auxiliary.